Solution of this logarithmic equation

Question

How to solve this equation $\log_{\sqrt{5}}x.\sqrt{\log_x{5\sqrt{5}}+\log_{\sqrt{5}}{5\sqrt{5}}}=-\sqrt{6}$ I dont know where to start this equation from. Converting log into powers even is not possible. Please help.

Answer 1

Hint

If you set $y=\ln(x)$ and use the fact that $\ln_a(b)=\frac{\ln(b)}{\ln(a)}$ your equation become $$\frac{2y}{\ln(5)}\sqrt{\frac{3/2\ln(5)}{y}+3}=-\sqrt 6$$