Same Log Both Sides

In $\log_{4}\left(3x-1\right) = \log_{4}\left(x+5\right)$ there is a log on each side, with the same base. Two logs of the same base are equal only when their insides are equal, because each output of a log comes from exactly one input. So drop the logs: $3x-1 = x+5$, which gives x = 3.

This needs the same base on both sides: $\log_{2}\left(x\right)$ and $\log_{4}\left(x\right)$ are different functions, so their insides cannot simply be matched.

Then check, as always. With x = 3 both insides are 8, which is positive, so the answer stands. If a candidate made an inside zero or negative, that log would have no value and the candidate would be thrown out.