Equation with logarithms with different bases.

Question

Solve the equation $$\log_x(2)\cdot \log_{2x}(2)=\log_{4x}(2)$$ for $x$.

I am not able to work out what I need to do. Can we find the answer using quadratic formula?

Answer 1

$\log_{x}{2}\cdot\log_{2x}{2}=\log_{4x}{2}$

$\frac{\log_{n}{2}}{\log_{n}{x}}\cdot\frac{\log_{n}{2}}{\log_{n}{2}+\log_{n}{x}}=\frac{\log_{n}{2}}{2\log_{n}{2}+\log_{n}{x}}$

$(\log_{n}{x})^{2}=2(\log_{n}{2})^{2}$

$\log_{n}{x}=\pm\sqrt{2}\log_{n}{2}$

$x=2^{\pm\sqrt{2}}$