Solve 4^r=1/8

$$4 ^ { r } = \frac { 1 } { 8 }$$

Solve for r

$$r = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Steps Using Definition of Logarithm

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

$$4^{r}=\frac{1}{8}$$

Take the logarithm of both sides of the equation.

$$\log(4^{r})=\log(\frac{1}{8})$$

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

$$r\log(4)=\log(\frac{1}{8})$$

Divide both sides by \log(4).

$$r=\frac{\log(\frac{1}{8})}{\log(4)}$$

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

$$r=\log_{4}\left(\frac{1}{8}\right)$$

Solve for r (complex solution)

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

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