Solve 5^3x-2=9

$$5 ^ { 3 x - 2 } = 9$$

Solve for x

$$x = \frac{2 {(\log_{5} {(3)} + 1)}}{3} \approx 1.121737463$$

Steps Using Definition of Logarithm

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

$$5^{3x-2}=9$$

Take the logarithm of both sides of the equation.

$$\log(5^{3x-2})=\log(9)$$

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

$$\left(3x-2\right)\log(5)=\log(9)$$

Divide both sides by \log(5).

$$3x-2=\frac{\log(9)}{\log(5)}$$

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

$$3x-2=\log_{5}\left(9\right)$$

Add 2 to both sides of the equation.

$$3x=2\log_{5}\left(3\right)-\left(-2\right)$$

Divide both sides by 3.

$$x=\frac{2\log_{5}\left(3\right)+2}{3}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3\ln(5)}+\frac{2\log_{5}\left(15\right)}{3}$$

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