Solve 2^2x+7=2^15

$$2 ^ { 2 x + 7 } = 2 ^ { 15 }$$

Solve for x

$$x=4$$

Steps Using Definition of Logarithm

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

$$2^{2x+7}=32768$$

Take the logarithm of both sides of the equation.

$$\log(2^{2x+7})=\log(32768)$$

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

$$\left(2x+7\right)\log(2)=\log(32768)$$

Divide both sides by \log(2).

$$2x+7=\frac{\log(32768)}{\log(2)}$$

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

$$2x+7=\log_{2}\left(32768\right)$$

Subtract 7 from both sides of the equation.

$$2x=15-7$$

Divide both sides by 2.

$$x=\frac{8}{2}$$

Solve for x (complex solution)

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

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