Solve 125^9x+2=150

$${ 125 }^{ 9x+2 } = 150$$

Solve for x

$$x=\frac{\log_{5}\left(6\right)-4}{27}\approx -0.106915454$$

Steps Using Definition of Logarithm

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

$$125^{9x+2}=150$$

Take the logarithm of both sides of the equation.

$$\log(125^{9x+2})=\log(150)$$

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

$$\left(9x+2\right)\log(125)=\log(150)$$

Divide both sides by \log(125).

$$9x+2=\frac{\log(150)}{\log(125)}$$

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

$$9x+2=\log_{125}\left(150\right)$$

Subtract 2 from both sides of the equation.

$$9x=\frac{\log_{5}\left(150\right)}{3}-2$$

Divide both sides by 9.

$$x=\frac{\frac{\log_{5}\left(150\right)}{3}-2}{9}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{27\ln(5)}+\frac{\log_{5}\left(6\right)}{27}-\frac{4}{27}$$

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