Solve 16^x=512

$${ 16 }^{ x } =512$$

Solve for x

$$x = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

Steps Using Definition of Logarithm

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

$$16^{x}=512$$

Take the logarithm of both sides of the equation.

$$\log(16^{x})=\log(512)$$

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

$$x\log(16)=\log(512)$$

Divide both sides by \log(16).

$$x=\frac{\log(512)}{\log(16)}$$

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

$$x=\log_{16}\left(512\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2\ln(2)}+\frac{9}{4}$$

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