Solve 4^2x=1/64

$${ 4 }^{ 2x } = \frac{ 1 }{ 64 }$$

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.

$$4^{2x}=\frac{1}{64}$$

Take the logarithm of both sides of the equation.

$$\log(4^{2x})=\log(\frac{1}{64})$$

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

$$2x\log(4)=\log(\frac{1}{64})$$

Divide both sides by \log(4).

$$2x=\frac{\log(\frac{1}{64})}{\log(4)}$$

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

$$2x=\log_{4}\left(\frac{1}{64}\right)$$

Divide both sides by 2.

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

Solve for x (complex solution)

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

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