Solve 3^2x=7

$$3 ^ { 2 x } = 7$$

Solve for x

$$x=\frac{\log_{3}\left(7\right)}{2}\approx 0.885621875$$

Steps Using Definition of Logarithm

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

$$3^{2x}=7$$

Take the logarithm of both sides of the equation.

$$\log(3^{2x})=\log(7)$$

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

$$2x\log(3)=\log(7)$$

Divide both sides by \log(3).

$$2x=\frac{\log(7)}{\log(3)}$$

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

$$2x=\log_{3}\left(7\right)$$

Divide both sides by 2.

$$x=\frac{\log_{3}\left(7\right)}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(3)}+\frac{\log_{3}\left(7\right)}{2}$$

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