Solve 6^2x=5

$$6 ^ { 2 x } = 5$$

Solve for x

$$x=\frac{\log_{6}\left(5\right)}{2}\approx 0.449122201$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$6^{2x}=5$$

Take the logarithm of both sides of the equation.

$$\log(6^{2x})=\log(5)$$

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

$$2x\log(6)=\log(5)$$

Divide both sides by \log(6).

$$2x=\frac{\log(5)}{\log(6)}$$

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

$$2x=\log_{6}\left(5\right)$$

Divide both sides by 2.

$$x=\frac{\log_{6}\left(5\right)}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(6)}+\frac{\log_{6}\left(5\right)}{2}$$

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