Find an inverse for the function log |x|

Consider the function $f(x) = \log |x|$ for $x < 0$ . Prove that this function has an inverse, determine the domain of this inverse, and find a formula to compute the inverse $g(y)$ .

Proof. From the discussion on page 146 of Apostol we know that a function which is continuous and strictly monotonic on an interval $[a,b]$ has an inverse on $[a,b]$ . The function $f(x) = \log |x|$ is continuous and strictly monotonic on the negative real axis; therefore, it has an inverse. We know it is continuous since the log function is continuous on the positive real axis, and $|x| > 0$ for all $x$ , in particular, for all $x < 0$ . Furthermore, we know it is strictly monotonic since

$f'(x) = \frac{1}{x} < 0 \qquad \text{for all } x < 0.$

Therefore, $f(x) = \log |x|$ has an inverse for all $x < 0$ . The domain of this inverse is the range of $\log |x|$ which is all of $\mathbb{R}. \qquad \blacksquare$