Solve 3^2x-1=3^7

$$3 ^ { 2 x - 1 } = 3 ^ { 7 }$$

Solve for x

$$x=4$$

Steps Using Definition of Logarithm

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

$$3^{2x-1}=2187$$

Take the logarithm of both sides of the equation.

$$\log(3^{2x-1})=\log(2187)$$

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

$$\left(2x-1\right)\log(3)=\log(2187)$$

Divide both sides by \log(3).

$$2x-1=\frac{\log(2187)}{\log(3)}$$

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

$$2x-1=\log_{3}\left(2187\right)$$

Add 1 to both sides of the equation.

$$2x=7-\left(-1\right)$$

Divide both sides by 2.

$$x=\frac{8}{2}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(3)}+4$$

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