Solve 3^n+2+2^2=19

$$3 ^ { n + 2 } + 2 ^ { 2 } = 19$$

Solve for n

$$n=\log_{3}\left(5\right)-1\approx 0.464973521$$

Steps Using Definition of Logarithm

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

$$3^{n+2}+4=19$$

Subtract 4 from both sides of the equation.

$$3^{n+2}=15$$

Take the logarithm of both sides of the equation.

$$\log(3^{n+2})=\log(15)$$

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

$$\left(n+2\right)\log(3)=\log(15)$$

Divide both sides by \log(3).

$$n+2=\frac{\log(15)}{\log(3)}$$

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

$$n+2=\log_{3}\left(15\right)$$

Subtract 2 from both sides of the equation.

$$n=\log_{3}\left(15\right)-2$$