Solve 4^x-2=1/2

$$4 ^ { x - 2 } = \frac { 1 } { 2 }$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Definition of Logarithm

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

$$4^{x-2}=\frac{1}{2}$$

Take the logarithm of both sides of the equation.

$$\log(4^{x-2})=\log(\frac{1}{2})$$

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

$$\left(x-2\right)\log(4)=\log(\frac{1}{2})$$

Divide both sides by \log(4).

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

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

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

Add 2 to both sides of the equation.

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

Solve for x (complex solution)

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

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