Solve 4^x=1/32

$$4 ^ { x } = \frac { 1 } { 32 }$$

Solve for x

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

Steps Using Definition of Logarithm

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

$$4^{x}=\frac{1}{32}$$

Take the logarithm of both sides of the equation.

$$\log(4^{x})=\log(\frac{1}{32})$$

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

$$x\log(4)=\log(\frac{1}{32})$$

Divide both sides by \log(4).

$$x=\frac{\log(\frac{1}{32})}{\log(4)}$$

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

$$x=\log_{4}\left(\frac{1}{32}\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(2)}-\frac{5}{2}$$

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