Solve (1/2)^x=8

$$( \frac { 1 } { 2 } ) ^ { x } = 8$$

Solve for x

$$x=-3$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{2}\right)^{x}=8$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{2}\right)^{x})=\log(8)$$

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

$$x\log(\frac{1}{2})=\log(8)$$

Divide both sides by \log(\frac{1}{2}).

$$x=\frac{\log(8)}{\log(\frac{1}{2})}$$

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

$$x=\log_{\frac{1}{2}}\left(8\right)$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(2)}-3$$

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