Solve (0.25)^x-3=32

$$( 0.25 ) ^ { x - 3 } = 32$$

Solve for x

$$x=0.5$$

Steps Using Definition of Logarithm

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

$$0.25^{x-3}=32$$

Take the logarithm of both sides of the equation.

$$\log(0.25^{x-3})=\log(32)$$

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

$$\left(x-3\right)\log(0.25)=\log(32)$$

Divide both sides by \log(0.25).

$$x-3=\frac{\log(32)}{\log(0.25)}$$

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

$$x-3=\log_{0.25}\left(32\right)$$

Add 3 to both sides of the equation.

$$x=-\frac{5}{2}-\left(-3\right)$$

Solve for x (complex solution)

$$x=-\frac{i\pi n_{1}}{\ln(2)}+0.5$$

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