# 6.1.1 AP Calculus Inverse of e^x

• MHB
• karush
In summary: It was a typo that I made because I was focused on the original pdf. The title of the original pdf is "(e^x)^2". That's exactly the same as e^{2x}.It's understandable to make a typo when trying to explain a typo situation, but the best way to handle it is to correct it in a follow-up post.In summary, the inverse function of $f(x)=e^{2x}$ is incorrectly stated as $\dfrac{1}{2}\ln x$ in the given answer choices. The correct inverse function is $f^{-1}(x)=\ln\dfrac{x}{2}$.
karush
Gold Member
MHB
If $f^{-1}(x)$ is the inverse of $f(x)=e^{2x}$, 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...

Last edited:
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?

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 ...

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

karush said:
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!

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

HallsofIvy said:
Yes, but post 2 suggested that the original problem might be to find the inverse function of $$f(x)= e^{2x}$$, not of $$f(x)= x^{2x}$$ 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.

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

HallsofIvy said:
But i don't see anything in the first post that is connected with $$x^{2x}$$.

Ah yes. That's true. That was a typo when referring to a suggested possible typo about a typo in the opening post that was actually a presumed typo in the original pdf.

## What is the inverse of e^x in AP Calculus?

The inverse of e^x in AP Calculus is ln(x), also known as the natural logarithm. This means that if y = e^x, then x = ln(y).

## Why is the inverse of e^x important in AP Calculus?

The inverse of e^x is important in AP Calculus because it allows us to solve exponential equations and model exponential growth and decay in real-world situations. It also helps us to understand the relationship between exponential and logarithmic functions.

## How do you find the inverse of e^x in AP Calculus?

To find the inverse of e^x in AP Calculus, we can use the property that the inverse of a function switches the x and y values. So, for e^x, we can switch the x and y values and solve for y to get ln(x).

## What is the domain and range of the inverse of e^x in AP Calculus?

The domain of the inverse of e^x is all positive real numbers, since ln(x) is only defined for positive values of x. The range is all real numbers, since ln(x) can take on any real value.

## What is the graph of the inverse of e^x in AP Calculus?

The graph of the inverse of e^x, ln(x), is a logarithmic curve that approaches the x-axis as x approaches 0 and increases without bound as x increases. It has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

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