Solve 64^n=16

$$64 ^ { n } = 16$$

Solve for n

$$n=\frac{2}{3}\approx 0.666666667$$

Steps Using Definition of Logarithm

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

$$64^{n}=16$$

Take the logarithm of both sides of the equation.

$$\log(64^{n})=\log(16)$$

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

$$n\log(64)=\log(16)$$

Divide both sides by \log(64).

$$n=\frac{\log(16)}{\log(64)}$$

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

$$n=\log_{64}\left(16\right)$$

Solve for n (complex solution)

$$n=\frac{\pi n_{1}i}{3\ln(2)}+\frac{2}{3}$$

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