Solve 3(5^3x+1)=36

$$3 ( 5 ^ { 3 x + 1 } ) = 36$$

Solve for x

$$x=\frac{\log_{5}\left(12\right)-1}{3}\approx 0.18131977$$

Steps Using Definition of Logarithm

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

$$3\times 5^{3x+1}=36$$

Divide both sides by 3.

$$5^{3x+1}=12$$

Take the logarithm of both sides of the equation.

$$\log(5^{3x+1})=\log(12)$$

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

$$\left(3x+1\right)\log(5)=\log(12)$$

Divide both sides by \log(5).

$$3x+1=\frac{\log(12)}{\log(5)}$$

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

$$3x+1=\log_{5}\left(12\right)$$

Subtract 1 from both sides of the equation.

$$3x=\log_{5}\left(12\right)-1$$

Divide both sides by 3.

$$x=\frac{\log_{5}\left(12\right)-1}{3}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3\ln(5)}+\frac{\log_{5}\left(12\right)}{3}-\frac{1}{3}$$

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