Solve 6=(25/6)^2x

$$6 = ( \frac { 25 } { 6 } ) ^ { 2 x }$$

Solve for x

$$x=\frac{\log_{\frac{25}{6}}\left(6\right)}{2}\approx 0.627755215$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$\left(\frac{25}{6}\right)^{2x}=6$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{25}{6}\right)^{2x})=\log(6)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$2x\log(\frac{25}{6})=\log(6)$$

Divide both sides by \log(\frac{25}{6}).

$$2x=\frac{\log(6)}{\log(\frac{25}{6})}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$2x=\log_{\frac{25}{6}}\left(6\right)$$

Divide both sides by 2.

$$x=\frac{\ln(6)}{2\ln(\frac{25}{6})}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(\frac{25}{6})}+\frac{\log_{\frac{25}{6}}\left(6\right)}{2}$$

$$n_{1}\in \mathrm{Z}$$