Solve 4^x-3=32

$$4 ^ { x - 3 } = 32$$

Solve for x

$$x = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

Steps Using Definition of Logarithm

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

$$4^{x-3}=32$$

Take the logarithm of both sides of the equation.

$$\log(4^{x-3})=\log(32)$$

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

$$\left(x-3\right)\log(4)=\log(32)$$

Divide both sides by \log(4).

$$x-3=\frac{\log(32)}{\log(4)}$$

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

$$x-3=\log_{4}\left(32\right)$$

Add 3 to both sides of the equation.

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

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(2)}+\frac{11}{2}$$

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