Solve (4^-1)^x=64

$$( 4 ^ { - 1 } ) ^ { x } = 64$$

Solve for x

$$x=-3$$

Steps Using Definition of Logarithm

Calculate 4 to the power of -1 and get \frac{1}{4}.

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

Take the logarithm of both sides of the equation.

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

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

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

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

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

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

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

Solve for x (complex solution)

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

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