Solve 23^2x=18

$$23 ^ { 2 x } = 18$$

Solve for x

$$x=\frac{\log_{23}\left(18\right)}{2}\approx 0.460911671$$

Steps Using Definition of Logarithm

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

$$23^{2x}=18$$

Take the logarithm of both sides of the equation.

$$\log(23^{2x})=\log(18)$$

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

$$2x\log(23)=\log(18)$$

Divide both sides by \log(23).

$$2x=\frac{\log(18)}{\log(23)}$$

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

$$2x=\log_{23}\left(18\right)$$

Divide both sides by 2.

$$x=\frac{\log_{23}\left(18\right)}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(23)}+\frac{\log_{23}\left(18\right)}{2}$$

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