Solving exponential equations using logarithms (where two things are being multipled together)

Question

I am having trouble with the following problem, which is about solving exponential equations using logarithms with base 10:

Initially, I thought I'd take the log of $2^x-1$ and $4^{2x+1}$ separately, and then multiply them. But that didn't work, because there's no rule for multiplying logs to combine them. Then, I thought I'd multiply the base and exponents before converting them to a $\log$ - this got an exponent of $2x^2$, which seems to overcomplicate the problem. So now I don't know.

Does anyone have a solution to this problem? If so, I greatly appreciate it.

Answer 1

Notice $4^{2x+1} = 2^{2(2x+1)}$

$2^{x-1} \cdot 4^{2x+1} = 2^{x-1} \cdot 2^{2(2x+1)} = 2^{x-1 + 2(2x+1)} = 2^{5x+1}=32=2^5$ which implies $5x+1=5$ so $x = \frac{4}{5}$