Solve 5^x=256

$$5 ^ { x } = 256$$

Solve for x

$$x=8\log_{5}\left(2\right)\approx 3.445412465$$

Steps Using Definition of Logarithm

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

$$5^{x}=256$$

Take the logarithm of both sides of the equation.

$$\log(5^{x})=\log(256)$$

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

$$x\log(5)=\log(256)$$

Divide both sides by \log(5).

$$x=\frac{\log(256)}{\log(5)}$$

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

$$x=\log_{5}\left(256\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(5)}+8\log_{5}\left(2\right)$$

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