Remember, any $\log$ to the base $b$, such as $\log_b(y)$, is in its heart an exponent. Let us start, then, from your expression
Raise $b$ to the powers we see on each side. We get
The left-hand side simplifies greatly. We get
The rest is elementary algebra. The above equation is (for $x\ne 0$) equivalent to
which is an easily solved linear equation.
Comment: There was no need to do the preliminary manipulation. We are told that
Raise $b$ to the power on the left-hand side, the right-hand side. We obtain
By the "laws of logarithms" this yields