Solve 16=8^-x+5

$$16 = 8 ^ { - x + 5 }$$

Solve for x

$$x = \frac{11}{3} = 3\frac{2}{3} \approx 3.666666667$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$8^{-x+5}=16$$

Take the logarithm of both sides of the equation.

$$\log(8^{-x+5})=\log(16)$$

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

$$\left(-x+5\right)\log(8)=\log(16)$$

Divide both sides by \log(8).

$$-x+5=\frac{\log(16)}{\log(8)}$$

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

$$-x+5=\log_{8}\left(16\right)$$

Subtract 5 from both sides of the equation.

$$-x=\frac{4}{3}-5$$

Divide both sides by -1.

$$x=-\frac{\frac{11}{3}}{-1}$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\pi n_{1}}{3\ln(2)}+\frac{11}{3}$$

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