Solve (1/4)^2x=(1/2)^3

$$( \frac { 1 } { 4 } ) ^ { 2 x } = ( \frac { 1 } { 2 } ) ^ { 3 }$$

Solve for x

$$x=\frac{3}{4}=0.75$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{4}\right)^{2x}=\frac{1}{8}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{4}\right)^{2x})=\log(\frac{1}{8})$$

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

$$2x\log(\frac{1}{4})=\log(\frac{1}{8})$$

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

$$2x=\frac{\log(\frac{1}{8})}{\log(\frac{1}{4})}$$

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

$$2x=\log_{\frac{1}{4}}\left(\frac{1}{8}\right)$$

Divide both sides by 2.

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

Solve for x (complex solution)

$$x=-\frac{i\pi n_{1}}{2\ln(2)}+\frac{3}{4}$$

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