Solve 6^n=512

$$6 ^ { n } = 512$$

Solve for n

$$n=9\log_{6}\left(2\right)\approx 3.481675265$$

Steps Using Definition of Logarithm

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

$$6^{n}=512$$

Take the logarithm of both sides of the equation.

$$\log(6^{n})=\log(512)$$

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

$$n\log(6)=\log(512)$$

Divide both sides by \log(6).

$$n=\frac{\log(512)}{\log(6)}$$

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

$$n=\log_{6}\left(512\right)$$

Solve for n (complex solution)

$$n=\frac{2\pi n_{1}i}{\ln(6)}+9\log_{6}\left(2\right)$$

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