Solve 5^3x+1=25

$$5 ^ { 3 x + 1 } = 25$$

Solve for x

$$x=\frac{1}{3}\approx 0.333333333$$

Steps Using Definition of Logarithm

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

$$5^{3x+1}=25$$

Take the logarithm of both sides of the equation.

$$\log(5^{3x+1})=\log(25)$$

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

$$\left(3x+1\right)\log(5)=\log(25)$$

Divide both sides by \log(5).

$$3x+1=\frac{\log(25)}{\log(5)}$$

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

$$3x+1=\log_{5}\left(25\right)$$

Subtract 1 from both sides of the equation.

$$3x=2-1$$

Divide both sides by 3.

$$x=\frac{1}{3}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{3\ln(5)}+\frac{1}{3}$$

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