# Solution of this logarithmic equation

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.

• Welcome to Math.SE! Is the big square root inside or outside of the $\log_{\sqrt{5}}$? – Hrodelbert Sep 28 '15 at 12:52
• I have written the exact printed expression that i have, but I can say with utmost surety that it is outside, as the question will lose its symmetricity and a solution will be very difficult. – user274843 Sep 28 '15 at 12:59

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