Solve 8*31^x=80

$$8 \times 31 ^ { x } = 80$$

Solve for x

$$x=\log_{31}\left(10\right)\approx 0.670528152$$

Steps Using Definition of Logarithm

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

$$8\times 31^{x}=80$$

Divide both sides by 8.

$$31^{x}=10$$

Take the logarithm of both sides of the equation.

$$\log(31^{x})=\log(10)$$

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

$$x\log(31)=\log(10)$$

Divide both sides by \log(31).

$$x=\frac{\log(10)}{\log(31)}$$

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

$$x=\log_{31}\left(10\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(31)}+\log_{31}\left(10\right)$$

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