Solve 4^-2x=64

$$4 ^ { - 2 x } = 64$$

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.

$$4^{-2x}=64$$

Take the logarithm of both sides of the equation.

$$\log(4^{-2x})=\log(64)$$

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

$$-2x\log(4)=\log(64)$$

Divide both sides by \log(4).

$$-2x=\frac{\log(64)}{\log(4)}$$

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

$$-2x=\log_{4}\left(64\right)$$

Divide both sides by -2.

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

Solve for x (complex solution)

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

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