Solve (1/2)^y=4

$$( \frac { 1 } { 2 } ) ^ { y } = 4$$

Solve for y

$$y=-2$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{2}\right)^{y}=4$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{2}\right)^{y})=\log(4)$$

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

$$y\log(\frac{1}{2})=\log(4)$$

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

$$y=\frac{\log(4)}{\log(\frac{1}{2})}$$

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

$$y=\log_{\frac{1}{2}}\left(4\right)$$

Solve for y (complex solution)

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

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