Solve 592*3^2x=74

$$592 \cdot 3 ^ { 2 x } = 74$$

Solve for x

$$x=-\frac{3\log_{3}\left(2\right)}{2}\approx -0.94639463$$

Steps Using Definition of Logarithm

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

$$592\times 3^{2x}=74$$

Divide both sides by 592.

$$3^{2x}=\frac{1}{8}$$

Take the logarithm of both sides of the equation.

$$\log(3^{2x})=\log(\frac{1}{8})$$

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

$$2x\log(3)=\log(\frac{1}{8})$$

Divide both sides by \log(3).

$$2x=\frac{\log(\frac{1}{8})}{\log(3)}$$

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

$$2x=\log_{3}\left(\frac{1}{8}\right)$$

Divide both sides by 2.

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

Solve for x (complex solution)

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

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