Question

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

Answer 1

`x=5`

Explanation:

The function `f(x) = 10^x` is strictly monotonically increasing on its (implicit) domain `(-oo, oo)` with range `(0, oo)`.

Its inverse `f^(-1)(x) = log(x)` is strictly monotonically increasing on its (implicit) domain `(0, oo)` with range `(-oo, oo)`.

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]}