Question

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

Answer 1

`f^(-1)(x)=ln(x)`

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.