Question

What is the inverse of f(x)=e^x

Answer 1

Answer:

$f^{-1}(x)=\ln{x}$

Step-by-step explanation:

An inverse function is any function that "undoes" another function. If we think of the function $f$ as some kind of machine that takes in a number $x$ as input and produces a number $f(x)$ as an output, when we give inverse function $f^{-1}$ the number $f(x)$ as in input, we get $x$, our original input, as the output $f^{-1}(x)$

We need a function that undoes $f(x)=e^x$, and the natural choice for undoing an exponent is with a logarithm. Here, our base is $e$, so we'll choose $\log_e{x}=\ln{x}$ as our inverse function. Let's see how that works:

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

$\ln{e^x}$ is the power we have to raise $e$ to to get $e^x$, which is $x$, so

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

And we have our function.