Solve 6^2x-2=485

$$6 ^ { 2 x - 2 } = 485$$

Solve for x

$$x = \frac{\log_{6} {(485)} + 2}{2} \approx 2.725719606$$

Steps Using Definition of Logarithm

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

$$6^{2x-2}=485$$

Take the logarithm of both sides of the equation.

$$\log(6^{2x-2})=\log(485)$$

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

$$\left(2x-2\right)\log(6)=\log(485)$$

Divide both sides by \log(6).

$$2x-2=\frac{\log(485)}{\log(6)}$$

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

$$2x-2=\log_{6}\left(485\right)$$

Add 2 to both sides of the equation.

$$2x=\log_{6}\left(485\right)-\left(-2\right)$$

Divide both sides by 2.

$$x=\frac{\log_{6}\left(485\right)+2}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(6)}+\frac{\log_{6}\left(485\right)}{2}+1$$

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