# Thread: 202 AP Calculus Inverse of e^x

1. If $f^{-1}(x)$ is the inverse of $f(x)=e^x$, then $f^{-1}(x)=$

$a. \ln\dfrac{2}{x}$
$b. \ln \dfrac{x}{2}$
$c. \dfrac{1}{2}\ln x$
$d. \sqrt{\ln x}$
$e. \ln(2-x)$

ok, it looks slam dunk but also kinda ???

my initial step was
$y=e^x$ inverse $\displaystyle x=e^y$
isolate
$\ln{x} = y$

the overleaf pdf of this project is here .... lots of placeholders...

2.

3. The inverse of $f(x)=e^x$ is $f^{-1}(x) = \ln{x}$

... there is an obvious mistake in the answer choices.

Maybe a typo? $f(x) = e^{2x}$ ???

well graphing it looks like its (c)

so how???

5. the graph is close, but no cigar.

$f(1)=e \implies f^{-1}(e) =1$

however, if $f^{-1}(x)=\dfrac{1}{2}\ln{x}$, then $f^{-1}(e) = \dfrac{1}{2} \ne 1$

have another look ...

ok looks like your suggestion of $y=x^{2x}$ is correct

7. Originally Posted by karush
ok looks like your suggestion of $y=x^{2x}$ is correct
And that was not what he suggested! Please be more careful what you are writing or you are just wasting our time!

post #2 looks like a suggestion to me!

9. Yes, but post 2 suggested that the original problem might be to find the inverse function of $\displaystyle f(x)= e^{2x}$, not of $\displaystyle f(x)= x^{2x}$ as you say in post 5!

10. Originally Posted by HallsofIvy
Yes, but post 2 suggested that the original problem might be to find the inverse function of $\displaystyle f(x)= e^{2x}$, not of $\displaystyle f(x)= x^{2x}$ as you say in post 5!
I inspected the pdf. It looks to me that the typo is in the original problem.
That is, I think the writers of the pdf made the mistake.
We can only guess about what it should have been.

11. But i don't see anything in the first post that is connected with $\displaystyle x^{2x}$.