Solve 8^5x=3

$$8 ^ { 5 x } = 3$$

Solve for x

$$x=\frac{\log_{2}\left(3\right)}{15}\approx 0.105664167$$

Steps Using Definition of Logarithm

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

$$8^{5x}=3$$

Take the logarithm of both sides of the equation.

$$\log(8^{5x})=\log(3)$$

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

$$5x\log(8)=\log(3)$$

Divide both sides by \log(8).

$$5x=\frac{\log(3)}{\log(8)}$$

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

$$5x=\log_{8}\left(3\right)$$

Divide both sides by 5.

$$x=\frac{\log_{2}\left(3\right)}{3\times 5}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{15\ln(2)}+\frac{\log_{2}\left(3\right)}{15}$$

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