Solve 16^x-7=19

$$16 ^ { x - 7 } = 19$$

Solve for x

$$x = \frac{\log_{2} {(19)} + 28}{4} \approx 8.061981878$$

Steps Using Definition of Logarithm

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

$$16^{x-7}=19$$

Take the logarithm of both sides of the equation.

$$\log(16^{x-7})=\log(19)$$

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

$$\left(x-7\right)\log(16)=\log(19)$$

Divide both sides by \log(16).

$$x-7=\frac{\log(19)}{\log(16)}$$

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

$$x-7=\log_{16}\left(19\right)$$

Add 7 to both sides of the equation.

$$x=\frac{\log_{2}\left(19\right)}{4}-\left(-7\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2\ln(2)}+\frac{\log_{2}\left(19\right)}{4}+7$$

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