Solve 1/3=31^n

$$\frac { 1 } { 3 } = 31 ^ { n }$$

Solve for n

$$n=-\log_{31}\left(3\right)\approx -0.319923233$$

Steps Using Definition of Logarithm

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

$$31^{n}=\frac{1}{3}$$

Take the logarithm of both sides of the equation.

$$\log(31^{n})=\log(\frac{1}{3})$$

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

$$n\log(31)=\log(\frac{1}{3})$$

Divide both sides by \log(31).

$$n=\frac{\log(\frac{1}{3})}{\log(31)}$$

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

$$n=\log_{31}\left(\frac{1}{3}\right)$$