How to solve this logarithmic equation through algebraic manipulation? I was working through an Art of Problem Solving workbook when I encountered a very frustrating problem.
Solve the equation $\log_{2x}216=x$, where $x\in \mathbb{R}$.
I understand how to find the answer by inspection (all you have to do is stare at it for a few seconds), but I'm trying to figure out how to solve it algebraically. Every time I try to manipulate the problem, I either find myself running in circles or I create some convoluted expression that's even harder to deal with than the original problem. Forgive me if this is a very easy question - I'm just completely lost here. Thanks!

In terms of using inspection, I think this makes the inspection easier.
$$\ln_{2x}216=x \Rightarrow \ln 6^3 = x \ln 2x \Rightarrow 3 \ln 6 = x \ln2x $$
If you guess that the linear factors are equal ($x=3$) you immediately see the logarithmic factors are equal.
