Solve 4=16^x+3

$$4 = 16 ^ { x + 3 }$$

Solve for x

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

Steps Using Definition of Logarithm

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

$$16^{x+3}=4$$

Take the logarithm of both sides of the equation.

$$\log(16^{x+3})=\log(4)$$

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

$$\left(x+3\right)\log(16)=\log(4)$$

Divide both sides by \log(16).

$$x+3=\frac{\log(4)}{\log(16)}$$

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

$$x+3=\log_{16}\left(4\right)$$

Subtract 3 from both sides of the equation.

$$x=\frac{1}{2}-3$$

Solve for x (complex solution)

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

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