Solve 16^-b=4^3

$$16 ^ { - b } = 4 ^ { 3 }$$

Solve for b

$$b = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Steps Using Definition of Logarithm

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

$$16^{-b}=64$$

Take the logarithm of both sides of the equation.

$$\log(16^{-b})=\log(64)$$

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

$$-b\log(16)=\log(64)$$

Divide both sides by \log(16).

$$-b=\frac{\log(64)}{\log(16)}$$

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

$$-b=\log_{16}\left(64\right)$$

Divide both sides by -1.

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