Solve 11^5x=33

$$11 ^ { 5 x } = 33$$

Solve for x

$$x=\frac{\log_{11}\left(33\right)}{5}\approx 0.291631382$$

Steps Using Definition of Logarithm

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

$$11^{5x}=33$$

Take the logarithm of both sides of the equation.

$$\log(11^{5x})=\log(33)$$

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

$$5x\log(11)=\log(33)$$

Divide both sides by \log(11).

$$5x=\frac{\log(33)}{\log(11)}$$

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

$$5x=\log_{11}\left(33\right)$$

Divide both sides by 5.

$$x=\frac{\log_{11}\left(33\right)}{5}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{5\ln(11)}+\frac{\log_{11}\left(33\right)}{5}$$

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