Solve 4^a+6-5=13

$${ 4 }^{ a+6 } -5 = 13$$

Solve for a

$$a=\log_{2}\left(3\right)-\frac{11}{2}\approx -3.915037499$$

Steps Using Definition of Logarithm

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

$$4^{a+6}-5=13$$

Add 5 to both sides of the equation.

$$4^{a+6}=18$$

Take the logarithm of both sides of the equation.

$$\log(4^{a+6})=\log(18)$$

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

$$\left(a+6\right)\log(4)=\log(18)$$

Divide both sides by \log(4).

$$a+6=\frac{\log(18)}{\log(4)}$$

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

$$a+6=\log_{4}\left(18\right)$$

Subtract 6 from both sides of the equation.

$$a=\frac{\log_{2}\left(18\right)}{2}-6$$