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analisi

matematica

lezione

.ii

00:00:06

so let's continue the lesson from where we

00:00:09

left off

00:00:12

we saw first the discussion

00:00:14

on natural powers I

00:00:17

focused above all on the question of

00:00:19

inactivity on the activities of

00:00:21

possible invertibility so we

00:00:24

said we said this figure

00:00:27

summarizes let's say the conclusion of

00:00:29

this discussion that for the

00:00:31

odd powers that you find in the figure on the very

00:00:34

right

00:00:36

the function is objective therefore vertible

00:00:39

just as for the function it is found on the

00:00:43

left that the even power

00:00:45

then you have to restrict the domain to 0

00:00:48

plus infinite and the condominium to 0 plus it is

00:00:50

finite then this becomes

00:00:51

objective so being objective

00:00:53

now we can define the

00:00:54

inverse functions we will have to pay

00:00:56

attention again however due to the effect of this

00:00:58

this figure that you see to distinguish

00:01:02

various powers the display powers

00:01:05

but what is the nth root function

00:01:08

it is nothing other than the

00:01:12

inverse function right to n when we can

00:01:15

define it then we need to do certain

00:01:17

operations we saw before then

00:01:20

based on the previous discussion we have

00:01:23

that it can be defined by let's draw

00:01:26

a nice table and distinguish even

00:01:28

and odd n then perennial paris and I want to

00:01:33

talk about inverse functions we

00:01:34

said that we need to restrict the

00:01:36

zeroinfinity 0 plus infinite so

00:01:39

when it passed to the inverse function I have to

00:01:41

exchange the domain domain so I

00:01:43

apparently don't notice anything because the

00:01:46

domain eco from mino are the same

00:01:49

in particular but perennially even as

00:01:52

defined the inverse function right to n

00:01:55

you have to take the usual general recipe

00:01:58

we saw a lesson previous

00:02:01

probably the 4 or 5 now 9 short

00:02:07

which is the following therefore it is the function

00:02:09

that associates to each y the only solution

00:02:12

is greater than or equal to zero because if

00:02:16

there are not two solutions but

00:02:18

always the positive ones there is only one

00:02:20

unique solution ex greater than zero dx

00:02:23

an is equal to y so for example if

00:02:28

you take one the nth root of one

00:02:32

based on this definition is that

00:02:34

only positive number which when raised to the

00:02:37

nv gives one therefore it would be one of

00:02:40

Chinese but of one to name one on this side

00:02:45

perennial odd instead for odd nei

00:02:47

I don't have to take any precautions

00:02:49

because the exponentiation function to

00:02:51

odd powers was already ticketed as a

00:02:53

function from r to r when the

00:02:55

inverse function passes I have to exchange domain

00:02:58

codomain but again I don't notice

00:03:00

anything because they are the themselves and the

00:03:04

definition is apparently the same

00:03:07

as on here but you have it in parallel

00:03:09

compare them now it is the only solution

00:03:12

is belonging to r I don't have to make

00:03:15

further restrictions to have

00:03:16

only one because I have seen that when the

00:03:19

power is odd this equation

00:03:21

actually due to the strict

00:03:23

monotonicity on the entire real axis with

00:03:25

a single solution

00:03:28

okay so let's see what

00:03:33

happens then both nth

00:03:35

root functions are strictly

00:03:37

increasing this is a general principle

00:03:40

when you invert a

00:03:42

strictly increasing function the console

00:03:44

inverses the same monotonicity therefore is

00:03:46

strictly chosen

00:03:48

if instead it had been the

00:03:49

strictly decreasing function its inverse

00:03:51

would have been strictly decreasing

00:03:54

the perennial nth root function of

00:03:59

perennial disparity is a function of

00:04:01

shots this too can be done easily but

00:04:05

we will probably see it in the graph both of the

00:04:07

most easily

00:04:08

this however in general principle the

00:04:10

inverse function of a function of

00:04:13

disappearing a function of shots for the

00:04:16

nth root with l even the question

00:04:18

doesn't have however I didn't put two bars

00:04:21

because it doesn't make sense it doesn't make sense

00:04:24

to deal with parity in the sense

00:04:28

we saw in the lesson review of

00:04:31

a function defined only on the

00:04:33

positives because if you remember I gave you

00:04:34

the premise to say that a

00:04:37

function couple of shots first of all the

00:04:39

function must be defined on a

00:04:40

set symmetric with respect to the origin

00:04:43

this function defined only on the

00:04:45

positives therefore it doesn't really make sense

00:04:47

no you cannot make the nth root

00:04:51

minus one because it is not defined as perennial

00:04:56

even so let's say this, this whole

00:05:00

discussion I'm having should also

00:05:02

already make it clear to you that when you

00:05:05

have a square root with a

00:05:08

certain expression underneath this square root

00:05:11

will be well defined if this expression

00:05:13

it must be positive is the reason he

00:05:16

explained it here is because otherwise the

00:05:18

inverse function root square

00:05:21

root fourth root sixth root is not

00:05:24

well defined ok so be careful

00:05:28

because in this first in this first

00:05:31

slot here it is first of all declared

00:05:35

what the domain is of definition of the

00:05:38

nth root if n is odd and all r

00:05:41

is even and only zeroinfinity

00:05:46

then enough with the properties that I wanted

00:05:49

that I wanted to highlight for this

00:05:52

function we use the

00:05:55

classical notation for the nth root that

00:05:57

this one here I always use the variable y to

00:06:02

remember which is an inverse function of

00:06:05

something and so I don't confuse

00:06:06

y

00:06:08

and often the notation

00:06:12

this here will also be used to indicate the root

00:06:14

the nth is also written as a

00:06:16

rational power xy to the 1 over an which is convenient for

00:06:20

many calculations and is also convenient to

00:06:22

do things that we will see in a

00:06:24

moment so for example the

00:06:27

square root you can also write as y

00:06:29

at half an

00:06:32

hour here are the graphs tell me

00:06:34

where the professor got them from I got

00:06:36

these graphs

00:06:38

out how I got you explained in

00:06:42

lesson 5 therefore they were obtained with

00:06:45

the procedure of turning the sheet of paper 90

00:06:47

degrees counterclockwise and then flipping

00:06:50

the axis of the

00:06:59

to these

00:07:02

graphs here if you now and rotate the

00:07:04

screen or more precisely perhaps it is

00:07:08

better if instead of rotating the screen

00:07:09

which then perhaps you are on your mobile phone it is

00:07:11

easily done but you are on the computer

00:07:13

risking breaking everything you would

00:07:16

perhaps be better off turning your head 90 degrees

00:07:18

clockwise twisted and then do this

00:07:23

mental operation of flipping

00:07:25

invert the y axis so that it is

00:07:28

directed in the right way found

00:07:30

exactly these graphs ok or in other

00:07:34

words sense do the reverse operation

00:07:36

on these puny flips you have to

00:07:38

find the right graph to the fourth and to

00:07:41

hicks respectively it's fine

00:07:46

so I explained this procedure to you

00:07:47

in diction 5 in particular

00:07:50

you see these are increasing monotonic functions

00:07:54

and also note that I can

00:07:57

already anticipate this you see that in the functions

00:08:01

before inverting it near the origin

00:08:04

the function tended to flatten out to

00:08:06

what we will see later on it

00:08:08

is called the horizontal tangent no the

00:08:11

horizontal tangent if you make

00:08:14

90 degree turns it becomes a

00:08:16

vertical tangent and in fact this explains

00:08:20

why in the graph there would appear to be

00:08:22

very steep slopes of the

00:08:26

vertical slopes at

00:08:27

the 'origin

00:08:30

then let's now look at the definition

00:08:31

of integer power and then this

00:08:35

will allow us to understand how

00:08:36

powers and negative power functions are defined

00:08:38

so now I take n more

00:08:42

generally belonging to the integers I want to

00:08:45

define the hicks function at n then

00:08:47

if n is positive I've done it before and I'm

00:08:51

fine if n is negative then ex an

00:08:56

I define it as one on hicks to the minus n pay

00:09:02

attention now minus n don't be

00:09:04

fooled by the minus sign

00:09:05

since n is negative minus n is positive

00:09:09

so essentially it changes you

00:09:12

put a minus sign in front of

00:09:13

the exponent and you take the fight

00:09:18

downstairs said fo so hug

00:09:21

now I'll give you an example that is perhaps

00:09:23

clearer for example the hicks function

00:09:25

to the minus two is

00:09:27

defined as one over hicks

00:09:29

squared and in general we will always use

00:09:33

these these these this how to

00:09:37

say this notation in the room less alpha

00:09:40

will be if I can always see it as a

00:09:42

swing salloukh then the fact that I

00:09:47

define it like this I have never declared to myself

00:09:50

what is the domain on which

00:09:52

this function is defined now I can say it

00:09:54

since it cannot be divided by 0 0

00:09:57

this is clear

00:10:00

when n is negative as in this

00:10:02

example here the example of n

00:10:04

2 the elevation function the n is defined as

00:10:07

its r minus the origin therefore I have to

00:10:10

remove the zero point because if not there

00:10:13

the function is not well defined ok one

00:10:17

divided by zero it cannot be done then from the

00:10:22

point of view of parity that

00:10:25

property this function is quite

00:10:27

easy exactly as before

00:10:29

when n is even whether it is positive or

00:10:31

negative it does not matter therefore even minus 2

00:10:34

for example, the function in the room m is even and

00:10:37

therefore follows this terminology

00:10:39

when n is an odd integer and an is

00:10:43

odd

00:10:45

as regards monotonicity

00:10:47

we must be careful because now I

00:10:50

will show you on the graph

00:10:51

the hicks elevation to n with negative n

00:10:59

is not never nor globally on its domain

00:11:04

and definition it is never nor

00:11:06

increasingly increasing but always has

00:11:09

two intervals of monotonicity then I

00:11:13

wrote it here when n less than or equal to

00:11:15

one the function in the room n is monotonic

00:11:18

strictly decreasing for its

00:11:20

positives

00:11:22

this which an whether it is even or odd

00:11:26

this is quite easy to see

00:11:28

because let's remember that perennial less than or

00:11:31

equal to minus 1 we have defined the

00:11:34

hall n as one on the hall minus n this

00:11:37

exponent minus mrs positive and therefore

00:11:41

when they are in the positive numbers the more

00:11:44

hicks becomes bigger and the more hicks alla is

00:11:47

minus n becomes large because I saw

00:11:50

before that the

00:11:53

natural and monotonically increasing exponentiation on the

00:11:57

positives on the positive semi-axis but the more

00:12:01

this becomes the larger the denominator

00:12:02

becomes and the smaller the result of

00:12:05

the whole fraction will be therefore the

00:12:08

more hicks it becomes larger and the smaller the

00:12:09

final result

00:12:10

aimed at by this function

00:12:12

therefore this tells me that the function

00:12:14

strictly decreasing on positives we

00:12:20

also observe that room n

00:12:22

always for negative n and therefore smaller

00:12:24

white minute tends to take on

00:12:31

increasingly larger values as it

00:12:34

approaches zero I see it from here

00:12:36

I always see it from this writing because

00:12:39

when hicks approaches 0x to the minus n

00:12:42

minus n positive remember start and

00:12:46

sisters is a number that is becoming

00:12:48

smaller and smaller positive so

00:12:52

this dividing 1 by an increasingly

00:12:54

smaller number which is like say that I am

00:12:56

multiplying 1 by an increasingly larger number

00:12:58

and therefore this

00:13:00

fraction resulted when I get closer to the right it is

00:13:03

said from zero it is becoming

00:13:06

larger and larger it tends to more infinity in fact

00:13:09

this function the hicks motion to an

00:13:12

with enel and negative is not limited

00:13:14

above his SUV is more infinite

00:13:18

vice versa instead when extend and I

00:13:22

put it in quotation marks because we have never

00:13:24

defined this concept yet

00:13:26

we will do it in the next few weeks it has more

00:13:28

infinity

00:13:29

the result of division 1 on the room

00:13:32

minus n approaches zero because here

00:13:34

now the denominator is becoming

00:13:36

more and more enormous and therefore you do one in

00:13:38

a million 1 on the billiards 1 in 100,000

00:13:40

billion

00:13:41

the final number is smaller and smaller it

00:13:44

is approaching 0 to 0 without ever

00:13:48

reaching it it is this function one on

00:13:50

island minus n can never be equal to

00:13:53

zero because it is never possible to obtain

00:13:55

0

00:13:56

as a result of dividing one

00:13:58

by something

00:14:00

this should be viviano then I

00:14:02

will schematically draw you

00:14:04

approximate graphs then from here you see

00:14:08

this graph and it is symmetrical with respect to the

00:14:11

origin so this is the case of

00:14:14

gunshots in fact this is the graph of

00:14:15

room minus 1 to 1 on hicks

00:14:18

you see that in the origin they are what

00:14:21

are called what is called

00:14:23

vertical asymptote between instead when

00:14:27

hicks becomes larger and larger the values

00:14:29

assumed a function get closer and closer

00:14:30

to 0 without ever reaching them

00:14:33

then this these the graph of

00:14:35

this function the one on the left

00:14:38

should clarify why I said that

00:14:42

this function is not globally

00:14:44

monotonic to this function the

00:14:49

function on the left is monotonically

00:14:51

decreasing for positive hicks and is

00:14:54

monotonic decreasing for negative hicks

00:14:56

but globally despite the

00:14:59

monotonicity intervals the type of monotonicity if the

00:15:00

same it is not true that it is monotonic no it

00:15:05

is not true that it is monotonous decreasing because

00:15:07

for example look look at what it is

00:15:10

worth in correspondence with two and a half

00:15:13

type here it is a small let's say it appears

00:15:17

negative it takes on negative values

00:15:19

while for devices it takes on

00:15:21

values positive therefore in going from

00:15:23

negative to positive the values increase

00:15:25

but then afterwards they decrease therefore there are

00:15:30

two intervals of decreasing monotonicity

00:15:33

but overall I cannot say that the

00:15:35

function is monotonically decreasing the one

00:15:40

on the right instead the graph you see is

00:15:42

symmetrical with respect to the y axis

00:15:45

therefore it is the even case and for example

00:15:48

this the graph of a squared switch

00:15:53

then observe that in the case of an

00:15:57

even exponent

00:15:58

the function being even cannot be

00:16:00

objective again this can also be seen

00:16:03

from the graph in the case of an

00:16:06

odd exponent for example salami 1

00:16:09

this and tickets goes you can also see from the

00:16:11

graph if I consider it how it works from

00:16:15

r minus 0 to values in d minus 0 in other

00:16:17

words the image of the hicks function

00:16:20

at least n with

00:16:25

with n natural odd is r minus

00:16:31

the origin then this is a function

00:16:33

objectively I wonder would you know how to find the

00:16:35

inverse function at this point you don't

00:16:37

have to go very far and it's just to

00:16:39

use the definition of inverse functions

00:16:41

and see if you can solve this

00:16:44

famous equation 1 suisse equals y

00:16:48

by solving this equation with respect to

00:16:50

if you will find the expression of the

00:16:52

function inverse is fine then

00:16:56

once we have also observed the

00:16:59

rational powers let's excuse the

00:17:02

inter powers and let's move on let's talk about

00:17:04

power and rationals so the aim here

00:17:08

is to be able to define more or

00:17:11

less sense that has a minimum of sense

00:17:14

let's say what it means to do hicks

00:17:17

to the 7 thirds for example what it means to

00:17:20

do while it is quite banal of hicks

00:17:23

to the seventh we know how to say xxx seven

00:17:26

times on the a7 thirds however we don't

00:17:29

really know no what it means I do hicks

00:17:31

maybe same 7 thirds times it doesn't add up

00:17:35

so much no then let's define a

00:17:40

national power therefore I take p and q two

00:17:43

natural numbers different from zero and without

00:17:45

common divisors and therefore the fraction p

00:17:47

on which it will already be reduced will already be reduced

00:17:50

to the minimum terms by one as

00:17:53

he said then I define the function which

00:17:56

is the stone shaykh salah p on which then I

00:17:59

will tell you what the domain is because he will have

00:18:01

discussed for the moment I will tell you

00:18:03

operationally how this

00:18:05

operation is defined the power p on cohesis but how

00:18:10

then xp on which this is the symbol and on

00:18:12

this side of the equal I will tell you how it is

00:18:15

manufactured it is manufactured first doing hicks

00:18:18

to the api which is the numerator of the

00:18:22

fraction and then doing everything to the 1 on

00:18:25

which the professor says however

00:18:28

we have not defined what is at work on

00:18:30

cu of course we have defined

00:18:32

it we have defined

00:18:36

the elevation exactly here a power 1 juice is a

00:18:38

notation to say the cohesive root but ok

00:18:43

so what is written here how do you

00:18:45

do the power for example 7

00:18:48

thirds right is done in the seventh room and then

00:18:52

left is the cube root because one

00:18:54

third is the cube root so in other

00:18:57

words hicks all'api su cu you see it

00:19:00

as the composition of the two functions

00:19:02

icsa nappi and the room 1 sukuk is the root

00:19:05

almost but of composition is opposition

00:19:10

see lesson maybe four because

00:19:14

originally I had to do it in

00:19:16

lesson 4 but then now that I think about it maybe

00:19:18

we did a lesson 5 after

00:19:21

possibly correct control so

00:19:23

you have the right reference that therefore you

00:19:28

have to compose these two functions

00:19:30

then you have to be very very careful I have

00:19:32

n't told you yet where this function is defined

00:19:37

the reason is because it is not trivial because

00:19:40

we must remember that the domain of

00:19:42

definition of the root quesi ma

00:19:43

depends on the parity of q, that is, it depends

00:19:46

on dry, whether it is divisible by two or not

00:19:48

as we saw before, so since

00:19:50

in the definition of the power piscu

00:19:52

a root quasi ma enters and then

00:19:55

we will have to be very careful

00:20:00

for understand the domain definition pixar

00:20:02

up on which we must distinguish various cases

00:20:04

first case the even numerator is the

00:20:09

denominator and odd then in this

00:20:13

case the function hicks to 1 on which

00:20:15

therefore the root coheses but with qt you shoot

00:20:19

and it is finished on all r we said no

00:20:21

so the operation that I have to do does

00:20:25

n't need any precautions whatsoever I

00:20:27

can do it whatever the hicks are I

00:20:30

take hicks defined on rxh b it is what it

00:20:34

is now we will specify it in a moment

00:20:37

in reality that something better than

00:20:38

what it is

00:20:39

and then I make it the root almost but so much so

00:20:42

odd this is always well

00:20:43

defined

00:20:44

so when the denominator

00:20:46

of the exponent is odd and the numerator

00:20:52

seems to be of an odd woman the function

00:20:56

defined on all r where does it come in to

00:20:59

give us something more the parity of

00:21:01

numerator we observe that since

00:21:03

pies are even

00:21:05

then we saw in the

00:21:07

previous discussion that the hall always gives you an

00:21:09

even number so when I do hicks at b and

00:21:14

then I take the cohesive root

00:21:16

Czech mallardi th of the number even and always even

00:21:21

sorry for the cohesive root but you I wanted

00:21:24

to say I'm a little tired maybe the

00:21:27

cohesive root but of a positive number it is a

00:21:30

positive number so when pies are even and

00:21:36

odd the icsa lucky function on

00:21:39

how to define r on everything and it

00:21:41

always takes positive values

00:21:43

furthermore it is an even function

00:21:45

this will tell you I leave it to be verified it is not

00:21:48

difficult basically just use the

00:21:50

fact that hicks to the eagle an even portion

00:21:55

we ask for another case p and odd eq and

00:21:59

odd now actually here from the point of

00:22:02

view in the domain of the definition there is

00:22:04

nothing there is not nothing new

00:22:08

because pixels at 1 on which the root

00:22:11

question is defined on all r so

00:22:13

here too the definition domain of all r of

00:22:16

this operation that I want to do is

00:22:17

defined on all r so from a certain point of

00:22:22

view the definition domain as before

00:22:24

if the Russell two names I love and odd

00:22:26

the function and it is over all the

00:22:29

fact that there is the shots

00:22:31

we see that it gives us some other

00:22:32

properties since p and q are

00:22:35

odd then when I do the

00:22:39

root the power more less hicks ok

00:22:44

this by definition of power of on

00:22:46

which I have to do less hicks at the api and then

00:22:48

roots quesi but ok but p and odd therefore

00:22:54

less hicks at the fright less in the room p

00:22:57

less hicks in brackets all the b is

00:22:59

equal to less hicks at the b because

00:23:01

exponentiation feet but it is a

00:23:04

portion of shots or on the other hand also

00:23:11

the cohesive root but with which odd is a

00:23:15

function of spal and we observed it

00:23:16

before so this minus that you have here

00:23:18

you can take it further out and

00:23:21

this now in the room p at 1 on which is

00:23:23

precisely the plus power on which so

00:23:25

now look only at the first term is

00:23:27

the last one you have shown that fd minus

00:23:29

hicks equals minus fds

00:23:31

so this case is a

00:23:33

function of shots so for example

00:23:35

case that falls into this case here

00:23:38

hicks at 7 thirds 7 odd and 3 odd

00:23:41

hicks to the 7 thirds and a function to

00:23:43

come on all r that is odd and

00:23:46

furthermore it is positive for positive hicks and

00:23:50

is negative for negative xvi what other

00:23:55

case we have left we have left in which the

00:23:57

case in where the denominator is even and you

00:24:02

will tell me then there are two cases left

00:24:03

because the numerator should be

00:24:05

even or odd no because I told you

00:24:07

that pi is closed with no common divisors

00:24:09

therefore if pi is even q cannot be even

00:24:11

we have already excluded it here

00:24:13

only the case remains in which p is odd and

00:24:15

paris then here we must be very

00:24:17

careful because in this case the

00:24:20

function in the root coheses but since two

00:24:22

even and it is finite only for positives

00:24:26

therefore the operation that I want to do is

00:24:29

defined only if hicks to the api it is

00:24:32

a positive number

00:24:35

ok but p is odd so when is

00:24:41

the exponentiation odd gives me

00:24:43

a positive result only when

00:24:46

hicks is positive so in other words

00:24:49

now I have to be careful because

00:24:50

when I take an exponentiation I

00:24:53

divide like hicks to the five quarters

00:24:56

the fact that there is that quarters therefore

00:24:59

that there is a middle fourth root

00:25:01

means that this function is defined only for

00:25:03

positive numbers

00:25:11

this one now that I wrote to you here

00:25:15

forget it completely

00:25:17

obviously this too was a

00:25:18

remnant of a remnant state this is

00:25:23

the thing from before that I copied and copied and

00:25:25

pasted it and it has nothing to do with it

00:25:29

as I told you before the function

00:25:33

is now defined only for positive hicks

00:25:37

so it makes no sense to ask the question

00:25:39

of parity disparity ok okay so

00:25:46

let's see if there is 'is no let's move

00:25:48

directly to the irrational power

00:25:50

after ok okay so you have to be very

00:25:56

careful then in summary having to

00:25:59

go to the bone when you do the

00:26:02

hicks function raised to a

00:26:05

national number p on cu what you have to

00:26:07

pay attention to above all is if the

00:26:10

denominator of this fraction is even or

00:26:13

odd ok that it separates the function will be

00:26:16

defined only for positive hicks if it is

00:26:19

odd the function will be defined on

00:26:21

all r okay let's see

00:26:28

ok we are yes we still have some

00:26:30

time to do what we want to do

00:26:35

now we have nothing left to do a type of

00:26:37

power farming which I don't know

00:26:40

if you have ever thought about,

00:26:43

we saw in some last lesson

00:26:46

that there are numbers that are not

00:26:50

rationed, rational numbers do not

00:26:52

exhaust the entire range of numbers that

00:26:55

interest us, in fact we

00:26:58

really want to work with numbers real ones that

00:27:00

we have seen that have particular free taxes

00:27:02

such as the axiom of continuity

00:27:04

which is very important to demonstrate

00:27:06

the existence of the upper and

00:27:08

lower bounds so then

00:27:11

the question arises for example if I take an

00:27:13

irrational number that cannot be

00:27:15

written like a powerful like a

00:27:18

fraction what does it mean to do hicks

00:27:21

at the root of 2 for example what an

00:27:23

operation how do I define it then the

00:27:25

question is very delicate in reality

00:27:28

so now here I give you a

00:27:30

definition for why because we are

00:27:37

teaching a lesson we want to put pen to paper

00:27:39

white of the things of the formulas we want

00:27:42

to do something that is at least

00:27:43

complete so I'll tell you something and

00:27:45

then obviously in reality there

00:27:49

would be a bit of work to understand

00:27:51

this definition that I'm giving you now

00:27:54

and to make it work

00:27:56

so let's say this slide is just

00:27:58

to tell you that hicks at the root of 2

00:28:01

can be defined is not a problem but there is

00:28:04

work to be done this work we

00:28:07

will not do it it is not the purpose of this course

00:28:11

to enter into these meanders let's say lido

00:28:15

just some small food for

00:28:17

thought to understand what here's

00:28:20

how to define this type

00:28:23

of function then let's briefly discuss

00:28:25

the alpha room function where alpha and

00:28:29

belongs to remember this symbol

00:28:32

this yes but it means r minus q therefore it is

00:28:35

a number that belongs to r but does not

00:28:37

belong to qu therefore it is a number that

00:28:40

is not represented as piscu therefore

00:28:43

all the effort I have made so far to

00:28:45

construct these powers even with

00:28:48

rational exponent but it is apparently useless

00:28:52

then to define what ykk is on

00:28:54

alpha and I can't dance like before I say

00:28:58

okay I'll do hicks all 'bees that

00:29:16

in lesson

00:29:20

3 in the density property of q in r

00:29:25

this property told me that there are

00:29:29

real numbers that cannot be

00:29:31

written as reactionary numbers but

00:29:34

each of these numbers can be

00:29:36

approximated with arbitrary precision

00:29:40

so you take the

00:29:42

irrational number and decide how much it is the

00:29:44

maximum error you want to commit and

00:29:46

you will find a rational that approximates

00:29:50

this irrational number with an error

00:29:52

which is the one you have chosen therefore

00:29:56

close in an arbitrary way close to

00:29:58

any irrational they found

00:30:00

rational hand and then this fact here

00:30:04

combined with the fact that we have already having made the

00:30:06

effort to define what it means

00:30:09

to national power it essentially

00:30:12

tells us that okay it won't be impossible

00:30:15

what we have to do

00:30:17

then the definition that I propose

00:30:20

is this one then first of all for

00:30:26

every hicks greater than or equal to zero

00:30:28

because when alpha nor rational therefore

00:30:31

non-rational power breeding

00:30:34

is defined only and well defined

00:30:37

only for hicks greater than or equal to

00:30:40

zero conventionally then if we want we

00:30:42

can come to an agreement and also make some

00:30:44

but for us it will always be like this

00:30:47

so for example iss to the root of 2

00:30:49

it will be a well-defined function only

00:30:52

on the positives and

00:30:56

we define this function in this way therefore

00:30:57

we define it starting from the

00:31:00

rational power breeding

00:31:03

we support here all these

00:31:05

increasingly general power constructions

00:31:07

always rely on the previous ones

00:31:09

so when hicks is between 0 and 1

00:31:13

I take the 'inferior right to the api on

00:31:17

cu when this varies what is this

00:31:22

is the inferior infimum of the set of

00:31:25

numbers of this form

00:31:34

which are strictly less

00:31:36

than this alpha ok then briefly

00:31:41

because I take the lower bound of

00:31:43

these numbers

00:31:44

because when hicks is between 0 and 1

00:31:49

then this quantity here this

00:31:53

quantity here is very na decreasing to

00:31:58

decreasing with respect to the exponent that is

00:32:00

more than I take exponents large ones

00:32:03

that are approaching the malfa is more the

00:32:05

result of this little number here is

00:32:06

small so we are basically I'm

00:32:11

taking a kind of limit I do

00:32:14

n't want to call it that because I

00:32:16

haven't yet defined limits what

00:32:17

they are but the idea is that you say words

00:32:22

I fix my x301 I do hicks at the

00:32:29

api on cu hits on cu

00:32:31

national number smaller than others this I know

00:32:33

what it means and then I try to

00:32:35

bring this national number here

00:32:38

on which more and more to alpha to this

00:32:41

operation of yes it gets closer and closer

00:32:43

to a limit point, that is, we get closer to

00:32:46

something which is precisely the

00:32:47

lower extremum, therefore the largest of the

00:32:52

largest of the smallest of these together

00:32:55

and this by definition on alpha

00:32:58

obviously I understand that it may seem like

00:33:01

a construction bit bizarre but it

00:33:08

can hardly be done simpler

00:33:12

for hicks is instead strictly

00:33:14

larger than one I take something completely

00:33:16

similar only now I have to take

00:33:18

the upper bound because these numbers

00:33:20

when the base hicks is larger than one

00:33:23

are instead monotonically increasing with respect

00:33:26

to the exponent but the idea is always the

00:33:28

same I take my fixed ex because

00:33:32

I want I want to define how much the

00:33:33

function is worth in the alpha room at the point it it

00:33:35

slow fixed I take hicks at the api juice I know

00:33:38

that as defined I do in the p room and then

00:33:41

I extract the cohesin root and then I try

00:33:43

to bring this rational p su cu closer

00:33:45

and closer to this

00:33:48

irrational alpha this procedure already a

00:33:51

terminal point that at the

00:33:53

upper extreme of these numbers and and okay

00:33:57

we have defined we have defined a

00:34:00

big word because you take this

00:34:03

definition as good but then we need to

00:34:05

demonstrate in short that this info

00:34:08

this on papiano certain properties well

00:34:10

anyway let's say so much so as not to have

00:34:13

an idea trying to give you an idea of

00:34:15

what the difficulties are in defining the

00:34:17

on alpha with irrational alpha and how they

00:34:20

can be circumvented I would say that we have it

00:34:22

wasted too much time let's make a

00:34:24

final reminder about the powers therefore

00:34:28

power calculation rules based

00:34:30

on the definition of the power function with

00:34:32

the exponent whatever you prefer rational integer

00:34:36

[Music]

00:34:37

natural or real you have these two

00:34:41

rules then when I say based on the

00:34:44

definition means that precisely

00:34:50

from the construction of the functions that I

00:34:52

have given you it can be demonstrated that

00:34:55

these properties are valid as it, let's not

00:34:58

waste time doing these

00:34:59

demonstrations because really if we can't

00:35:01

get it anymore let's keep it in mind though because

00:35:04

they are very important therefore

00:35:06

calculation rules and the powers hicks to the alpha

00:35:08

brics to the beta equals icsa the alpha

00:35:10

beta so when you multiply 2

00:35:15

exponentiations with the same base you

00:35:18

keep the base you add and add

00:35:20

the exponents this little rule must

00:35:22

certainly be clear to you up to the

00:35:24

hicks all school 'alpha to beta is instead

00:35:28

written as hicks to al faber meta

00:35:32

in the end due to national exponents

00:35:34

actually showing these properties

00:35:36

starting from the fairly

00:35:38

easy definition for irrational exponents

00:35:41

especially with the definition that I

00:35:42

gave you a little more complicated let's take it

00:35:45

for good let's be satisfied and we don't

00:35:47

even have to kill ourselves with the effort to

00:35:50

really demonstrate everything otherwise we wo

00:35:53

n't finish him

00:35:55

we must also take into account that we are

00:35:57

engineers so then it's a small

00:36:04

example a small summary example

00:36:06

just to make you familiar I

00:36:08

've put this example for you those who are

00:36:10

better than you come true a

00:36:11

stupid example I'll do it because I know from

00:36:14

experience again that there is often a

00:36:16

good percentage of the students who

00:36:19

in front of this thing here

00:36:22

this multiplication which I then write on

00:36:24

the blackboard which does this remain

00:36:27

completely dumbfounded c o god what

00:36:29

happened it happened that I

00:36:32

simply used the properties of the powers

00:36:34

this one then hicks right root and the

00:36:37

comic salon script half 16 teams the

00:36:40

script as hicks to minus two

00:36:43

I use this property therefore this icsa

00:36:46

not half for i will be minus 2 I write it

00:36:48

like the salons half minus two

00:36:51

one half minus 2 I do the necessary

00:36:53

algebraic operations I can write it

00:36:55

as one minus 4 everything really everything

00:36:57

over 214 is minus 3 is here

00:37:01

high level mathematics is hicks to the negative three means

00:37:06

it's like saying the 16 room three means because

00:37:10

the minus sign I told you before is like

00:37:11

saying one on hicks that power deprived

00:37:14

of the minus sign and hicks to the three means

00:37:18

we saw in the section in the

00:37:21

previous section is that of the

00:37:24

rational powers and how to say hicks to the cube

00:37:28

extracted square root therefore final

00:37:31

of this operation 1 on the

00:37:34

square root of cubed hicks which I could

00:37:36

also leave written as one in the room

00:37:38

three means or even as in the room minus three

00:37:40

months

00:37:43

very well then we are the exponentials

00:37:46

let's see how much time we have ok time is

00:37:49

starting to run out

00:37:52

so what is it the

00:37:53

exponential function the exponential function

00:37:55

once we have broken our

00:37:57

back doing the power function it is

00:38:00

quite easy to understand you

00:38:02

just have to be careful not to get too

00:38:04

messy now first of all a

00:38:06

main ingredient of the

00:38:08

exponential function is a basis what is a

00:38:10

base is a strictly positive number

00:38:12

other than one ok for example a half or

00:38:18

two are two possible bases

00:38:22

the base exponential function is defined as

00:38:24

the function defined on r with values in 0

00:38:28

plus infinity the because with value zero plus

00:38:31

infinity why this function define

00:38:34

in this way that for each real hicks

00:38:37

you associate with that the base which is fixed

00:38:41

raised to this exponent hicks which is

00:38:43

mobile ok then why because I

00:38:48

put 0 plus infinity in the condominium here

00:38:50

for a very simple reason

00:38:53

it has a base and positive therefore a

00:38:56

positive number raised to whatever you

00:38:59

want at this point because we have

00:39:02

really defined exponentiation with any

00:39:04

possible rubbish that has always been a positive number

00:39:06

which however is never 0 at the aics a

00:39:10

number raised to something can never

00:39:12

be zero therefore this explains why I

00:39:14

excluded zero then be careful

00:39:18

not to confuse power and exponentials

00:39:22

you will tell me professor why should we

00:39:24

make this type of confusion again

00:39:27

because I know that some students make

00:39:30

this type of confusion then in

00:39:33

powers be careful it is the base

00:39:36

which is variable and the fixed exponent

00:39:39

so do it for example hicks cubed

00:39:41

therefore the exponentiation hicks

00:39:44

the exponentiation cube is fixed the

00:39:48

base that went to arrive at the cube

00:39:50

instead moves varies in the exponentials

00:39:53

instead exactly the opposite

00:39:55

the base has been fixed for example 33 at

00:39:58

the aics therefore at base 3

00:40:02

the exponent to which I am going to raise this

00:40:04

3 is fixed instead it is variable

00:40:07

therefore being variable and being

00:40:10

real this I hope lets you

00:40:13

appreciate the reason why I first

00:40:16

took care to define what it means

00:40:20

say making a power with an

00:40:23

irrational exponent is because here when hicks

00:40:26

flows over the reals you will happen at a certain

00:40:28

point to make a to the root of 2 for

00:40:31

example now we know what it

00:40:34

means more or less at least we know that it

00:40:37

is well defined

00:40:40

then some properties the function

00:40:41

exponential of basis a is strictly

00:40:43

monotone therefore injective ok first

00:40:46

property that we like its image is

00:40:50

zeroinfinity therefore 0 infinity which I

00:40:52

put there a bit so in an innocent way it

00:40:54

is not just the condominium but it is precisely

00:40:56

the image this thing to demonstrate it

00:40:59

again more advanced tools will be needed

00:41:01

which will be the 3

00:41:03

of the intermediate values but for the moment

00:41:04

devil for good come on so both in

00:41:08

particular that the lower limit of the

00:41:12

of to the xv xvi belongs r is the

00:41:15

lower limit of the image which is zero

00:41:20

be careful that zero is excluded

00:41:24

from the image therefore 0 does not belong

00:41:26

to the image therefore we know that

00:41:28

when the lower bound is a certain

00:41:30

number which however does not belong

00:41:32

to the image then it means that that

00:41:34

set has no minimum therefore the

00:41:38

lower bound of the elevation is

00:41:40

lower bound of the exponential and zero but

00:41:43

zero is not the minimum of the function

00:41:45

because rightly as I said before there is no

00:41:47

hicks such that the aics

00:41:49

is zero therefore the exponential has no

00:41:53

minimum and only a lower bound which

00:41:55

has pads pay attention to the

00:42:01

upper extreme instead at the upper extreme

00:42:02

of the image which is more infinite here it is

00:42:05

therefore as a function from r in

00:42:08

infinite jargon it is va tickets which is on

00:42:12

reactivate as here I put

00:42:13

the image itself then it is injective

00:42:16

because strictly montorio then it

00:42:18

starts active invertible of its

00:42:20

inverse he talked about it in a moment here

00:42:25

now there is a question that I imagine

00:42:27

from the audience will be raised there and prof

00:42:30

but the lalla is strictly increasing

00:42:34

strictly of the recent who didn't

00:42:35

say it strictly monotonous and that's it not

00:42:39

the I said because it depends on the base a and

00:42:41

it sells a base in itself the base a is greater

00:42:46

than 1 then in this case it is easy

00:42:48

to see that the real number at the aics

00:42:50

grows as the exponent increases so

00:42:52

you have a number greater than one which

00:42:54

you raise to exponents increasingly larger

00:42:57

the result is always larger and

00:42:58

therefore with the base a greater than one

00:43:01

the elevation and sorry the exponential of the

00:43:04

base strictly increasing areas

00:43:07

the case has between 0 and 1 and a little more

00:43:10

important then in this case to

00:43:13

understand what is happening

00:43:14

instead of to the aics we write c1 its

00:43:17

the aics in this way as one its all

00:43:21

likes we can see so this is

00:43:24

always a consequence of the

00:43:27

power calculation rules from before because

00:43:29

I am using that one I can write it

00:43:31

as one the aics ok I now have one of its

00:43:40

since this time it is between 0 and 1

00:43:43

1 of its is a larger base than one which

00:43:47

if less than 1 1 of its ease greater than

00:43:49

therefore for the previous point

00:43:52

the exponential function with base 1 of its

00:43:56

is strictly increasing

00:43:59

in other words the function 1 its the aics

00:44:02

is strictly increasing but if one its

00:44:05

the strictly increasing aics then

00:44:07

it means that the strictly decreasing aics

00:44:13

ok I don't know if you agree about when

00:44:17

the monotonicity relation of the

00:44:18

inverts passed reciprocal that if this as

00:44:22

hicks increases the result of this division

00:44:24

of a work the aics becomes larger and

00:44:27

larger it means that it is the denominator

00:44:29

that is becoming smaller and smaller

00:44:31

so having said this here are the two graphs

00:44:37

here are the two graphs of the here are the two

00:44:45

graphs of the exponential function I

00:44:49

know what is happening there are some

00:44:50

machines that make noises ok

00:44:55

then in red the graph of the

00:44:58

basic exponential a when the basis is

00:45:00

greater than one you see strictly

00:45:03

increasing strictly increasing leaf 0

00:45:10

corresponds to taking ics that

00:45:12

go towards less infinite the more

00:45:14

infinite suv corresponds to sending ixfin

00:45:17

instead in black

00:45:20

you have the graph which is the typical one of

00:45:22

a strictly decreasing function

00:45:26

corresponding to taking a base

00:45:28

between 0 and 1 ok observe that

00:45:31

in any case the graph passes

00:45:33

from the point 0 to 1 because in correspondence

00:45:36

with zero you have at 0 which is one

00:45:38

we said it we declared it

00:45:39

at the beginning of this axis so this

00:45:42

nice section is fine then from the

00:45:47

power calculation rules

00:45:49

we write because we will need them we

00:45:51

automatically inherit the

00:45:54

calculation rules for the exponential no therefore a

00:45:56

to x1 to x2 this is the

00:45:59

multiplication of two numbers

00:46:00

the two power ligaments with the base

00:46:03

the same base and variable exponent and

00:46:06

we know from what was said before that

00:46:08

this does to x12 eastwood

00:46:13

calls this rule and 1

00:46:15

let's keep it in mind because I need it in

00:46:17

a moment and then there is also the rule and 2

00:46:20

which came from the second

00:46:22

power calculation rule that when I do

00:46:24

the aics to the alpha this is how to do

00:46:27

hara the alpha hicks whoever they are in the

00:46:30

country belonging to the real logarithms or

00:46:36

velho let me show you for a moment to

00:46:38

scare you then immediately remove many

00:46:41

immediately

00:46:42

then let's go back here for a moment

00:46:46

let's go back here for a moment this function is

00:46:50

objective we said the

00:46:52

basic exponential elevation to objective as

00:46:55

functions from r to 0 infinite if it is

00:46:58

objective I can define its

00:47:00

inverse function its inverse function is

00:47:02

exactly what is called the

00:47:05

logarithm based on so let's see if

00:47:07

we can understand the logarithms

00:47:09

based on well then let's take a

00:47:12

base again therefore a strictly

00:47:14

positive number different from one is defined as the

00:47:19

function logarithm and base as the

00:47:21

inverse function of the

00:47:23

basic exponential motion has therefore now

00:47:25

to understand why a lot insisted on a

00:47:28

general theoretical level on the

00:47:30

different functions because if you have

00:47:32

really understood them now you have no difficulty in

00:47:34

understanding the nth roots and the logarithm

00:47:37

is the inverse function of this function

00:47:41

so finally now we see

00:47:43

something interesting because since it must

00:47:46

be the inverse function of this

00:47:48

function

00:47:50

then first of all where is it defined and is it

00:47:54

over the codomain of the function that

00:47:56

was inverted therefore the

00:47:58

definition domain of the logarithm is zero plus

00:48:01

infinite what is the its image and its

00:48:05

image the domain

00:48:07

so I have to exchange my condominium node

00:48:09

as I always do

00:48:10

except that up until now the condominium often

00:48:13

coincided and no and I didn't notice

00:48:15

anything now domains and condominium

00:48:17

are different and when I exchange them I

00:48:19

notice something therefore based on the

00:48:21

general definition of inverse function

00:48:23

we saw in a last lesson we have

00:48:28

that this function is defined on 0 plus

00:48:31

infinity with values in r

00:48:33

and as defined and as you want it to be

00:48:36

defined and is finite as defined

00:48:38

any inverse function is the only

00:48:42

hicks solution belonging to r of y

00:48:46

equal to to the aics therefore logarithm

00:48:50

based on ypsilon is this sentence here

00:48:54

but if you that the palate and well

00:48:57

there will be no more mysteries about yuga rhythms ok

00:49:01

observation here I have already told you that they

00:49:03

put it in black and white the exponential

00:49:05

had domain r codomain 0 plus infinity

00:49:08

the logarithm with finite domain 0 and the

00:49:12

condo rr which they exchange and

00:49:13

finally we see something

00:49:15

interesting

00:49:16

now let's see if we have understood what

00:49:19

the logarithm is here now here I'll do some

00:49:22

exercises that I would be ashamed to do at

00:49:24

university though from experience

00:49:27

let's do them like this let's see if everything is

00:49:30

clear then let's take as the base a

00:49:32

equal to 2 logarithm to the base 2 of 4 what

00:49:37

it is by definition it is the only

00:49:40

former solution belonging to the reals I won't

00:49:44

write it down because there isn't

00:49:45

any anyway restriction see the real

00:49:49

the only hicks solution of the equation

00:49:52

two years ex equals 4 then when is for

00:49:56

which x2 raised the aics from four I would say it

00:49:59

is two no so logarithm and base 2

00:50:01

of four is two let's

00:50:04

do something a little more a little

00:50:06

more difficult logarithm to the base of a

00:50:08

half based on the definition is the only

00:50:12

hicks solution of two which is the basis to x1 which

00:50:17

a half for which x2 elevated vice is a

00:50:23

half a half I see it as two to the minus

00:50:26

12 to the minus one equals 2 the x6 equals

00:50:30

minus 1 therefore they are not even a

00:50:33

logarithm to the base two of 16 the only

00:50:35

solution xd 2 to the

00:50:46

based on two of 16 is

00:50:48

four

00:50:51

so for beyond 6 side let's try

00:50:55

to do even the most difficult case in which

00:50:56

we take a base smaller than them this

00:50:58

is a little more unusual

00:51:00

then we will see at the end of the lesson that there is no point in

00:51:03

busting your head with base in

00:51:06

geo million of one because there is a

00:51:07

trick to go from one to the other

00:51:09

but I won't tell you for the moment then

00:51:11

logarithm invaded a half of a half

00:51:15

the only solution xd a half to the AICS

00:51:18

a half to these its evil 1 therefore

00:51:21

logarithm on the basis of a half of a half

00:51:23

is one and this was really easy more

00:51:27

difficult logarithm on the basis here instead

00:51:29

there is a little mistake that I already highlight

00:51:33

logarithm on the basis of a half of 4

00:51:37

the only solution xd a half to the AICS

00:51:39

which 4 half the aics equals 44 what is it

00:51:47

and two to the second a half now I can

00:51:51

see it as two blades noix so with

00:51:54

that two to the minus hicks which a2 to the

00:51:56

second when minus which two so

00:51:58

when i up at least two if you don't believe it

00:52:01

starting from this which proves again

00:52:03

so put minus 2 here as an exponent

00:52:05

you see that gives it four logarithm to the

00:52:10

base of a half that I made a mistake in the two

00:52:12

a half I meant a quarter it's

00:52:14

the only solution xd a half to the AICS

00:52:17

equals a quarter

00:52:20

then a half times a half is or a quarter

00:52:22

so a half times a half that in

00:52:24

the middle of the second so the

00:52:27

logarithm solution to the base a half of a quarter

00:52:30

is two

00:52:31

ok we have seen a bit of an example at

00:52:34

this point so I'll put the graphs

00:52:38

again it says graphs commit them 6 facts

00:52:41

this is an inverse function once

00:52:44

I have made the graph of the function

00:52:46

to be inverted to make this graph I do the usual

00:52:48

operation I turn my

00:52:51

head 90 degrees or I turn the sheet of paper

00:52:53

90 degrees and then I invert the orientation

00:52:58

of the axis so you see in particular

00:53:02

that if the exponential the graphs

00:53:07

always passed from the point 0 1 because

00:53:10

the exponential of zero is worth one here

00:53:12

now the relationship is inverted and I

00:53:14

always go to point 10 that is in one all the

00:53:18

logarithms are worth zero logarithm based

00:53:21

on one equals zero because because the only

00:53:26

solution to the equation at the aics equals

00:53:29

one is its will raise again you see that

00:53:34

the red one corresponds to the base

00:53:37

greater than 1 and is increasing

00:53:40

when the base of the logarithm is less than

00:53:42

1 so as when the Basell

00:53:45

exponential less than one the

00:53:47

decreasing function very well

00:53:53

remembering that f composed of f minus 1 f

00:53:57

to minus one composed of f must give

00:53:58

the identity

00:53:59

then we have these two nice

00:54:02

relations that we should that we will use

00:54:04

often it is better for us to learn well therefore to

00:54:06

logarithm based on by

00:54:27

which I want to spend a few

00:54:29

moments are the calculation rules

00:54:34

for the logarithm then from the

00:54:36

calculation rules and 1 and 2 that we have seen for

00:54:39

the exponential and the definition of

00:54:41

inverse function we have the

00:54:44

corresponding calculation rules for the

00:54:46

logarithm which are these here l1 and l2

00:54:50

then I used the same numbering

00:54:52

to point out that 1 follows from the rule

00:54:56

and 1 and l 2 follows from the rule

00:54:59

then the first rule says that the sum

00:55:03

of the logarithms and the logarithm of the

00:55:06

product

00:55:07

the second says that the logarithm of an

00:55:10

elevation to power is equal to the

00:55:13

power multiplied by the logarithm

00:55:15

so in other words said said a bit

00:55:18

so vulgarly when

00:55:21

exponentiation within

00:55:23

the argument of the logarithm I take alpha and

00:55:25

bring it in front

00:55:29

then let's see the proof of the

00:55:31

first rule so that we can

00:55:33

check if we really have understood the

00:55:35

logarithm or not then based on the

00:55:40

rule and one for the exponentials I know

00:55:43

that when I write a to the logarithm day1

00:55:48

plus the logarithm based on y 2 you see

00:55:52

quartz has raised to a sum of two

00:55:55

numbers then I can

00:55:57

write this as the produced between two

00:55:59

power ligaments with the same base

00:56:02

which is independent of these two exponents lassie

00:56:06

on the other hand note well here it is

00:56:07

written a to the logarithm based on that which is

00:56:11

I am composing the exponential function

00:56:13

with the juice its inverse so this

00:56:15

must give me the identity i.e. y1 that you see

00:56:18

here here I'm doing the same has

00:56:22

composed with logarithm based on therefore

00:56:24

this must also give me identity therefore

00:56:27

based on this observation I can

00:56:29

I can write this identity to the

00:56:33

sum of these two logarithms and equals

00:56:35

y1 times y2 I can resell them this

00:56:40

trick of to the logarithm gives to

00:56:43

rewrite this product in yet another way

00:56:46

that is y1

00:57:05

two identities that I got so

00:57:08

this one here is this one

00:57:13

ok what did I get I got which has

00:57:17

to the sum of the logarithms and equal to the

00:57:22

logarithm of the product but now I use that

00:57:27

the exponential and injective so if

00:57:29

these two exponentials with these very

00:57:31

ugly arguments which are the

00:57:35

sum of the logarithms or the logarithm of the

00:57:37

product are equal this is possible

00:57:39

if only if the two exponents are

00:57:42

equal that is if the logarithm of the sum

00:57:45

equals the logarithm of the product sorry I

00:57:49

repeat it if the sum of the logarithms is

00:57:53

equal to the logarithm of the product ok

00:57:57

do it be very careful because the error tv

00:57:59

can do in this formula is that you

00:58:02

invert all the signs so many of

00:58:04

you will make the mistake of saying that

00:58:06

the logarithm of y1 plus y2 equals the

00:58:09

product delivered in this and very false

00:58:13

then another tried to demonstrate it

00:58:15

at home using exactly the same

00:58:17

type of order of ideas and instead of

00:58:21

using and one you will have to use the calculation rule

00:58:23

and two

00:58:26

then let's see we are almost done

00:58:29

saying our time I also wanted to do

00:58:32

an exercise on ugarits at this

00:58:33

point I don't I would say that there is no time for me

00:58:37

[Applause]

00:58:39

let me just say the last thing which is

00:58:41

the important thing about the change of base change of

00:58:45

base then it can happen in certain

00:58:47

situations that you have a logarithm in

00:58:49

a base that you don't like

00:58:51

for example logarithm of base 5 and you want to

00:58:55

rewrite it as a logarithm based on a

00:58:58

better base for you then we will see

00:59:00

later that our favorite base is a

00:59:02

mystical number which is called and

00:59:06

then there is a way to pass from one

00:59:08

phase to another it is the law of change

00:59:11

the base which is this logarithm here so

00:59:15

take two bases a and b logarithm in

00:59:18

base bd y is equal to the logarithm in base

00:59:21

b of a

00:59:22

so here you have the ratio between the two

00:59:24

bases

00:59:25

let's call it dear as base bta for

00:59:27

logarithm in base a of y so you see

00:59:30

that you can go from writing in

00:59:32

base b to that and base a there is a price

00:59:35

to pay which is this factor here the

00:59:39

logarithm and buzz emilia so let's see let's

00:59:41

conclude today's lesson with the

00:59:44

small demonstration of this fact

00:59:46

because the demonstration in itself is not

00:59:49

that it is important we won't lose

00:59:50

sleep but in reality it is a useful

00:59:53

new exercise to verify

00:59:55

logarithms 15

00:59:58

we need the demonstration as a check

01:00:01

to see if we have understood if we know how to

01:00:03

manipulate logarithms so let's not

01:00:06

subtract this type of exercise

01:00:08

this more than a trap proof

01:00:09

be an exercise

01:00:13

then the trick is exactly

01:00:16

what I used before the trick is to

01:00:19

write a number in this seemingly

01:00:23

abstruse form as a base and legal raised to

01:00:27

logarithm in that base of y

01:00:31

then if you use this trick when

01:00:34

you have logarithm based on b of y

01:00:36

this cute cute y here all by

01:00:39

itself you can try to write in this

01:00:42

completely idiotic form

01:00:44

it would seem no write it as a to the

01:00:47

logarithm base a of yes no

01:00:50

then why do you do this what do you

01:00:52

gain from it that now you have a logarithm

01:00:56

of a breeding to power and you know

01:01:00

how the logarithm works with respect to

01:01:02

breeding to power you take

01:01:04

the exponent and put it in front

01:01:09

that is if you use the calculation rule l2

01:01:12

which was this one is rightly

01:01:13

logarithm in base bd y to alpha equals

01:01:16

alpha logarithm in base a and b of y

01:01:19

dare use this formula with

01:01:22

ypsilon which will be this a and with

01:01:24

the exponent alpha this ugly gentleman

01:01:27

who is the logarithm based on

01:01:37

so and now pass

01:01:39

the exponent in front but this is

01:01:42

exactly I know it written

01:01:44

backwards but the product between two

01:01:45

real numbers and commutative so

01:01:48

you can adjust it exactly in order to

01:01:51

write it like this exactly the

01:01:55

formula we wanted then

01:01:56

particular case particular case taking

01:02:01

as a base b lin towards one of its ok so

01:02:08

if you have a base also one of its is a base

01:02:10

only what is the fact that if it has a base

01:02:12

greater than 1 1 its is a better base

01:02:15

than one and vice versa based on the

01:02:18

previous case I invite you to verifying it, however,

01:02:20

is quite simple because the

01:02:23

conversion factor from one base to another

01:02:25

here what you get is the

01:02:27

logarithm based on one suardi a which is

01:02:31

exactly minus one so you have this

01:02:34

nice relationship between for example if

01:02:37

you want to go from logarithm in base a

01:02:39

half logarithm of base 2 logarithm

01:02:42

invaded a half of ypsilon is equal to

01:02:44

minus the logarithm in base 2 of y

01:02:48

it's fine but I'll conclude this

01:02:51

lesson by leaving you this exercise

01:02:53

I had planned to do it together with you

01:02:56

if there wasn't time there is time for

01:03:00

today next time I would like

01:03:03

to concentrate on the

01:03:04

trigonometric functions which are very important

01:03:05

so let's do a nice thing

01:03:08

here now we won't discuss it together but I

01:03:12

'll leave it to you you will find it in the slides

01:03:16

available for the lesson there is the whole

01:03:18

process of

01:03:20

this it is a simple enough

01:03:24

logarithmic equation to solve

01:03:27

these taken from a task from a few

01:03:30

years ago the difficulty of the equation of

01:03:33

these motions to solve and that you have

01:03:35

two logarithms in two different bases however

01:03:37

with the base change formula that I

01:03:39

showed before it should not be

01:03:42

impossible to solve this exercise

01:03:44

approve c as I told you in the slides

01:03:47

that I make available to you you will find the

01:03:49

solution

01:03:50

if you can't do it there are no

01:03:53

steps that you don't understand as always I

01:03:55

invite you to write an email or

01:03:57

during the focus group to ask me

01:04:00

this exercise I wouldn't say

01:04:02

I'll see you at this point

01:04:04

next week on Wednesday

Description:

Prima parte della lezione 6 del corso di Analisi Matematica A, per il primo anno della Laurea Triennale in Ingegneria Meccanica (Università degli Studi di Ferrara), AA 2020/2021

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