Solve 16^n=32

$$16 ^ { n } = 32$$

Solve for n

$$n = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Steps Using Definition of Logarithm

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

$$16^{n}=32$$

Take the logarithm of both sides of the equation.

$$\log(16^{n})=\log(32)$$

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

$$n\log(16)=\log(32)$$

Divide both sides by \log(16).

$$n=\frac{\log(32)}{\log(16)}$$

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

$$n=\log_{16}\left(32\right)$$

Solve for n (complex solution)

$$n=\frac{\pi n_{1}i}{2\ln(2)}+\frac{5}{4}$$

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