How is the Logarithm derived from the exponential function? (aren't they inverses?) I've been learn

kreamykraka80 2022-07-10 Answered
How is the Logarithm derived from the exponential function? (aren't they inverses?)
I've been learning logs in school, and my teacher, friend, and I are stumped on something. How does one derive the logarithmic function from the exponential function? My friend thinks Tayler Series are the trick. Is he right? Is there a a better/simpler/more elegant way? Also, do calculators use taylor series to do logs? Thanks for the help
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Hayley Mccarthy
Answered 2022-07-11 Author has 19 answers
The answer (at least, one possible answer) is in your title! You can define logarithms as inverses of exponential functions.
However, this then prompts the question: how do you define the exponential function? Again there are various ways in which you could do this. One common way is to say that the exponential function f ( x ) = e x is the unique function which has the properties
d d x ( e x ) = e x and e 0 = 1   .
However, this raises some questions which are usually not answered (or worse, not even asked) in basic calculus courses. Here are two:
(1) How do we know that functions of the form a x are differentiable? After all, you will have met functions such as the absolute value which are not differentiable.
(2) Even if we assume that a x is differentiable, how do we know there is any value of a which makes its derivative the same function? After all, this is just asking us to find a by solving an equation, and there are many equations which have no solution, for example, a = a + 1
For these and other reasons it is often found better to do things the other way around: define the (natural) logarithm first by
ln x = 1 x d t t
for x > 0, and then define e x to be the inverse of ln x
It's a great question to think about and I hope this gives you a useful start.
A related question, also well worth thinking about: it's easy to say what we mean by π 2 , but what exactly do we mean by 2 π ?
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-04-02
Proof of a closed form of 01(lnx)ndx
asked 2022-11-19
Solve log 2 x = log 4 ( x + 6 ) for x using the change of base formula.
I already tried changing the base on both sides but that didn't work I know it must be in the form of a quadratic for a substitution to be made.
asked 2022-11-18
Find the solution to the differential equation
Assume x > 0 and let
x ( x + 1 ) d u d x = u 2 ,
u ( 1 ) = 4.
I started off by doing some algebra to get:
1 u 2 d u = 1 x 2 + x d x .
I then took the partial fraction of the right side of the equation:
1 u 2 d u = ( 1 x 1 x + 1 ) .
I then took the integral of both sides:
1 u = log x log ( x + 1 ) + C .
From here I don't know what to do because we are solving for u ( x ) and I'm not sure how to get that from 1 u
asked 2022-10-26
Evaluate log 64 using the change of base formula?
Is that even possible? I mean, there is no base.
asked 2022-05-24
Prove that a b c is a cube of some integer.
Given three integers a, b, and c such that a b + b c + c a is an integer too, prove that the product a b c is a cube.
asked 2022-05-14
Asymptotics of logarithms of functions
If I know that lim x f ( x ) g ( x ) = 1, does it follow that lim x log f ( x ) log g ( x ) = 1 as well? I see that this definitely doesn't hold for e f ( x ) e g ( x ) (take f ( x ) = x + 1 and g ( x ) = x), but I'm not sure how to handle the other direction.
asked 2022-04-01
How to solve
5log2(x3)=log2(x+1)