Solve 3=27^n

$$3 = 27 ^ { n }$$

Solve for n

$$n=\frac{1}{3}\approx 0.333333333$$

Steps Using Definition of Logarithm

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

$$27^{n}=3$$

Take the logarithm of both sides of the equation.

$$\log(27^{n})=\log(3)$$

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

$$n\log(27)=\log(3)$$

Divide both sides by \log(27).

$$n=\frac{\log(3)}{\log(27)}$$

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

$$n=\log_{27}\left(3\right)$$

Solve for n (complex solution)

$$n=\frac{2\pi n_{1}i}{3\ln(3)}+\frac{1}{3}$$

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