Solve 243^r=(1/3)

$$243 ^ { r } = ( \frac { 1 } { 3 } )$$

Solve for r

$$r=-\frac{1}{5}=-0.2$$

Steps Using Definition of Logarithm

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

$$243^{r}=\frac{1}{3}$$

Take the logarithm of both sides of the equation.

$$\log(243^{r})=\log(\frac{1}{3})$$

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

$$r\log(243)=\log(\frac{1}{3})$$

Divide both sides by \log(243).

$$r=\frac{\log(\frac{1}{3})}{\log(243)}$$

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

$$r=\log_{243}\left(\frac{1}{3}\right)$$

Solve for r (complex solution)

$$r=\frac{2\pi n_{1}i}{5\ln(3)}-\frac{1}{5}$$

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