Solve 20^-8v=35.5

$$20 ^ { - 8 v } = 35.5$$

Solve for v

$$v=\frac{\log_{20}\left(\frac{2}{71}\right)}{8}\approx -0.148942411$$

Steps Using Definition of Logarithm

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

$$20^{-8v}=35.5$$

Take the logarithm of both sides of the equation.

$$\log(20^{-8v})=\log(35.5)$$

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

$$-8v\log(20)=\log(35.5)$$

Divide both sides by \log(20).

$$-8v=\frac{\log(35.5)}{\log(20)}$$

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

$$-8v=\log_{20}\left(35.5\right)$$

Divide both sides by -8.

$$v=\frac{\ln(\frac{71}{2})}{-8\ln(20)}$$

Solve for v (complex solution)

$$v=-\frac{i\pi n_{1}}{4\ln(20)}+\frac{\log_{20}\left(\frac{2}{71}\right)}{8}$$

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