Solve 2^r=1/14

$$2 ^ { r } = \frac { 1 } { 14 }$$

Solve for r

$$r=-\log_{2}\left(14\right)\approx -3.807354922$$

Steps Using Definition of Logarithm

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

$$2^{r}=\frac{1}{14}$$

Take the logarithm of both sides of the equation.

$$\log(2^{r})=\log(\frac{1}{14})$$

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

$$r\log(2)=\log(\frac{1}{14})$$

Divide both sides by \log(2).

$$r=\frac{\log(\frac{1}{14})}{\log(2)}$$

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

$$r=\log_{2}\left(\frac{1}{14}\right)$$