Solve 16^a=1/32

$$16 ^ { a } = \frac { 1 } { 32 }$$

Solve for a

$$a = -\frac{5}{4} = -1\frac{1}{4} = -1.25$$

Steps Using Definition of Logarithm

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

$$16^{a}=\frac{1}{32}$$

Take the logarithm of both sides of the equation.

$$\log(16^{a})=\log(\frac{1}{32})$$

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

$$a\log(16)=\log(\frac{1}{32})$$

Divide both sides by \log(16).

$$a=\frac{\log(\frac{1}{32})}{\log(16)}$$

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

$$a=\log_{16}\left(\frac{1}{32}\right)$$