Solve 3^2x+1=243

$$3 ^ { 2 x + 1 } = 243$$

Solve for x

$$x=2$$

Steps Using Definition of Logarithm

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

$$3^{2x+1}=243$$

Take the logarithm of both sides of the equation.

$$\log(3^{2x+1})=\log(243)$$

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

$$\left(2x+1\right)\log(3)=\log(243)$$

Divide both sides by \log(3).

$$2x+1=\frac{\log(243)}{\log(3)}$$

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

$$2x+1=\log_{3}\left(243\right)$$

Subtract 1 from both sides of the equation.

$$2x=5-1$$

Divide both sides by 2.

$$x=\frac{4}{2}$$

Solve for x (complex solution)

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

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