Finding possible values for exponential identity without logarithms

Question

So I was teaching a student high school math and I came upon this problem:

Assuming one has more than oce solution for x, for the equation

$4^{ax} = 8.b^{x}$, find all possible solution of $a, b$.

Point is I can find the answer using trial and error, how can I algebraically solve it?

Thanks.

P.S. Using logarithms isn't allowed, only exponents!

Answer 1

if so we have $$2ax\ln(2)=3\ln(2)+x\ln(b)$$ and then: $$x(2a\ln(2)-\ln(3))=3\ln(2))$$ thus we get $$x=\frac{3\ln(2)}{2a\ln(2)-\ln(3)}$$ for $$b>0,2a\ln(2)-\ln(3)\ne 0$$