Solve 33^x=243

$$33 ^ { x } = 243$$

Solve for x

$$x=5\log_{33}\left(3\right)\approx 1.571013746$$

Steps Using Definition of Logarithm

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

$$33^{x}=243$$

Take the logarithm of both sides of the equation.

$$\log(33^{x})=\log(243)$$

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

$$x\log(33)=\log(243)$$

Divide both sides by \log(33).

$$x=\frac{\log(243)}{\log(33)}$$

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

$$x=\log_{33}\left(243\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(33)}+5\log_{33}\left(3\right)$$

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