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



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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

Answer & Explanation

Hayley Mccarthy

Hayley Mccarthy

Beginner2022-07-11Added 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 ?

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