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The solution set of $$(3x)^{\log 3} = (5x)^{\log 5}$$ is
Taking logarithms, $$\log 3(\log 3 + \log x) = \log 5(\log 5 + \log x)$$. So $$(\log 3)^2 - (\log 5)^2 = \log x(\log 5 - \log 3)$$, which gives $$\log x = -(\log 3 + \log 5) = -\log 15$$. Hence $$x = \frac{1}{15}$$ and the solution set is $$\left\{\frac{1}{15}\right\}$$.
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