6.3 The logarithm defined as an inverse function

$\sqrt{x}$ can be defined as the inverse function of $x^2$ . Recall the definition of an inverse function— $f^{-1}(x)$ is defined as the inverse of $f^1(x)$ if it reverses the inputs and outputs. So we can demonstrate this inverse relationship as follows:

Similarly, ${\text{log}}_{2}x$ is the inverse function of the exponential function $2^x$ .

(You may recall that during the discussion of inverse functions, $2^x$ was the only function you were given that you could not find the inverse of. Now you know!)

In fact, as we noted in the first chapter, $\sqrt{x}$ is not a perfect inverse of $x^2$ , since it does not work for negative numbers. $(-3{)}^2=9$ , but $\sqrt{9}$ is not $-3$ . Logarithms have no such limitation: ${\text{log}}_{2}x$ is a perfect inverse for $2^x$ .

The inverse of addition is subtraction. The inverse of multiplication is division. Why do exponents have two completely different kinds of inverses, roots and logarithms? Because exponents do not commute . $3^2$ and $2^3$ are not the same number. So the question “what number squared equals 10?” and the question “2 to what power equals 10?” are different questions, which we express as $\sqrt{\text{10}}$ and ${\text{log}}_2\text{10}$ , respectively, and they have different answers. $x^2$ and $2^x$ are not the same function, and they therefore have different inverse functions $\sqrt{x}$ and ${\text{log}}_2\text{10}$ .