# How do you solve log x = log 5 and find any extraneous solutions?

Jun 1, 2016

$x = 5$

#### Explanation:

The function $f \left(x\right) = {10}^{x}$ is strictly monotonically increasing on its (implicit) domain $\left(- \infty , \infty\right)$ with range $\left(0 , \infty\right)$.

Its inverse ${f}^{- 1} \left(x\right) = \log \left(x\right)$ is strictly monotonically increasing on its (implicit) domain $\left(0 , \infty\right)$ with range $\left(- \infty , \infty\right)$.

For each of these functions, since they are strictly monotonically increasing, they are also one-one.

So if $\log x = \log 5$ then $x = 5$.

graph{(y - 10^x)(x - 10^y) = 0 [-10, 10, -5, 5]}