Inverse functions and logarithms

Introduction

We learned recently about the exponential function. Now it’s time to learn about its inverse - the logarithm. We should talk more generally about inverse functions first.

Inverse functions

If $f:S\to T$ in one-to-one fashion, it’s inverse is the the function $f^{-1}:T\to S$ with the property that \[f^{-1}(f(x)) = x = f(f^{-1}(x)).\] I like to think of as if $f^{-1}$ “undoes” what the original function $f$ did.

Examples

Here are a few examples of inverse pairs.

$f(x) = x+1$ and $f^{-1}(x)=x-1$
$f(x) = 2x$ and $f^{-1}(x)=\frac{1}{2}x$
$f(x) = 2x+1$ and $f^{-1}(x)=\frac{1}{2}(x-1)$

Comments

$f^{-1}$ “undoes” what the function $f$ “does”.
- For this reason, inverse functions are useful for solving equations
Which function is $f$ and which is $f^{-1} is rather arbitrary.
- If $f^{-1}$ is the inverse of $f$, then $f$ is the inverse of $f^{-1}$.
An alternative formulation is that \[y=f(x) \text{ if and only if } x=f^{y}.\]

Mirror images

That last point is worth repeating \[y=f(x) \text{ if and only if } x=f^{y}.\]

As a result the roles of $x$ and $y$ flip.

Introduction