Solve 360=12^n

$$360 = 12 ^ { n }$$

Solve for n

$$n=\log_{12}\left(360\right)\approx 2.368742517$$

Steps Using Definition of Logarithm

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

$$12^{n}=360$$

Take the logarithm of both sides of the equation.

$$\log(12^{n})=\log(360)$$

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

$$n\log(12)=\log(360)$$

Divide both sides by \log(12).

$$n=\frac{\log(360)}{\log(12)}$$

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

$$n=\log_{12}\left(360\right)$$

Solve for n (complex solution)

$$n=\frac{2\pi n_{1}i}{\ln(12)}+\log_{12}\left(360\right)$$

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