Solve the equation : $9\log_5(x) = 25\log_x(5)$ . Express your answers in the form of $5^{\frac pq}$.
I started off by changing the base, as well as simplifying the right-hand side, and ended up with this: $$\log_5(x^9) = \frac{25}{\log_5(x)}$$ From that point I do know how to progress. I would put the $\log_5(x)$ to the other side, but multiplying logs here won't make it easier, so that idea is out the window. Changing the base to $x$ on the left-hand side would later give me an answer in terms of $x$ rather than in terms of $5$... Any help would be greatly appreciated!
we get $$9\log_5(x)=\frac{25}{\log_5(x)}$$ and we get $$\left(\log_5(x)\right)^2=\frac{25}{9}$$ can you proceed? taking the square root we get $$|\log_5(x)|=\frac{5}{3}$$