Explain why ln(x) is the inverse function to e^x?

Question

It's basically asking to prove that the ln(x) function does the inverse of e^x from a fundamental perspective. I'm not sure how to even approach the question, because it just seems so basic, so any help would be great, thanks.

Answer 1

Let y = e^x



Then swap x and y to find the inverse



Hence x = e^y



We need to take the logs of both sides to find y.



==> lnx = ln(e^y)



==> lnx = y



Hence y = lnx is the inverse of e^x

Answer 2

Inverse Of Ln

Answer 3

Usually, in Analysis, this follows by definition, ie by the way those functions are built in the first place.



Sometimes, one defines e^x, eg by its series or other means; then we note it is strictly increasing, so we can find an inverse function, and we note this inverse: ln(x), from which follows its usual properties.



But one can also *define* ln(x), eg by the integral of 1/t from 1 to x, for x>0; we then note it is strictly increasing, and we call its inverse function exp(x) ( or e^x ), from which follows its usual properties.



Of course, one could define ln(x) by a series, then e^x by a series, and later show they are the inverse of one another... But that would be necessarily complicated.



So it all depends how YOU encountered these functions, and if it was one of the two first ways, the answer is then basic as you said.



/Edit: a simpler way to state the above is this:

It makes not much sense to ask "why" ln(x) is the inverse of e^x, since it is *defined* that way, it is *constructed* that way.

That's the *fundamental* perspective.

Answer 4

The definition of an inverse function is that if you have a y as a function of x, then its inverse is x as a function of y. If y = x^2, then x = +/- sqrt(y).



The definitions of e^x and ln(x) are such that if y = e^x, then ln(y) = x.



The proof comes with the properties of the natural logarithm and the natural exponential:

if a = b, then ln a = ln b. so if y = e^x, then ln y = ln(e^x).

next, ln (a^b) = b ln a. So ln y = x ln e.

next, ln e = 1, so ln y = x (1), so ln y = x.