Solve (1/3)^r=1/27

$$( \frac { 1 } { 3 } ) ^ { r } = \frac { 1 } { 27 }$$

Solve for r

$$r=3$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{3}\right)^{r}=\frac{1}{27}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{3}\right)^{r})=\log(\frac{1}{27})$$

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

$$r\log(\frac{1}{3})=\log(\frac{1}{27})$$

Divide both sides by \log(\frac{1}{3}).

$$r=\frac{\log(\frac{1}{27})}{\log(\frac{1}{3})}$$

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

$$r=\log_{\frac{1}{3}}\left(\frac{1}{27}\right)$$