Solve 2.3^2x=18

$$2.3 ^ { 2 x } = 18$$

Solve for x

$$x = \frac{\log_{2.3} {(18)}}{2} \approx 1.735106315$$

Steps Using Definition of Logarithm

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

$$2.3^{2x}=18$$

Take the logarithm of both sides of the equation.

$$\log(2.3^{2x})=\log(18)$$

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

$$2x\log(2.3)=\log(18)$$

Divide both sides by \log(2.3).

$$2x=\frac{\log(18)}{\log(2.3)}$$

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

$$2x=\log_{2.3}\left(18\right)$$

Divide both sides by 2.

$$x=\frac{\ln(18)}{2\ln(\frac{23}{10})}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(2.3)}+\frac{\log_{2.3}\left(18\right)}{2}$$

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