Solve $\log_3(1/\sqrt{\log_3(x)})=\log_9(\log_9(x/3))$

Question

How can I solve the equation $$\log_3\left(\frac{1}{\sqrt{\log_3(x)}}\right)=\log_9\left(\log_9\left(\frac{x}{3}\right)\right)$$ I tried this $$\log_3(1)-\log_3\left(\sqrt{\log_3(x)}\right)=\log_9(\log_9(x)-\log_9(3))$$ $$-\log_3\left(\sqrt{\log_3(x)}\right)=\log_9(\log_9(x)-1/2)$$ $$-\log_3\left(\sqrt{\log_3(x)}\right)=\frac{1}{2}\log_3\left(\frac{1}{2}\log_3(x)-1/2\right)$$ But I can't continue.

Answer 1

We have that $$\begin{align}\log_3\left(\frac{1}{\sqrt{\log_3(x)}}\right)=\log_9\left(\log_9\left(\frac{x}{3}\right)\right)&\implies \ln\left(\sqrt{\frac{\ln3}{\ln x}}\right)=\frac{\ln\left(\ln\left(\frac{x}{3}\right)/\ln9\right)}{2}\\&\implies \sqrt{\frac{\ln3}{\ln x}}=\sqrt{\ln\left(\frac{x}{3}\right)/\ln9}\\&\implies \ln3\ln9=\ln x\ln\frac x3\end{align}$$ From which we see that an obvious solution is $x=9$.