Solve 32=16^x+1

$$32 = 16 ^ { x + 1 }$$

Solve for x

$$x=\frac{1}{4}=0.25$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$16^{x+1}=32$$

Take the logarithm of both sides of the equation.

$$\log(16^{x+1})=\log(32)$$

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

$$\left(x+1\right)\log(16)=\log(32)$$

Divide both sides by \log(16).

$$x+1=\frac{\log(32)}{\log(16)}$$

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

$$x+1=\log_{16}\left(32\right)$$

Subtract 1 from both sides of the equation.

$$x=\frac{5}{4}-1$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{2\ln(2)}+\frac{1}{4}$$

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