Solve =16*(1/4)^2x=32

$$= 16 \cdot ( \frac { 1 } { 4 } ) ^ { 2 x } = 32$$

Solve for x

$$x=-\frac{1}{4}=-0.25$$

Steps Using Definition of Logarithm

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

$$16\times \left(\frac{1}{4}\right)^{2x}=32$$

Divide both sides by 16.

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

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(\frac{1}{4}).

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

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

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

Divide both sides by 2.

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

Solve for x (complex solution)

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

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