Solve (1/5)^2-3x=25

$$( \frac { 1 } { 5 } ) ^ { 2 - 3 x } = 25$$

Solve for x

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{5}\right)^{-3x+2}=25$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{5}\right)^{-3x+2})=\log(25)$$

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

$$\left(-3x+2\right)\log(\frac{1}{5})=\log(25)$$

Divide both sides by \log(\frac{1}{5}).

$$-3x+2=\frac{\log(25)}{\log(\frac{1}{5})}$$

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

$$-3x+2=\log_{\frac{1}{5}}\left(25\right)$$

Subtract 2 from both sides of the equation.

$$-3x=-2-2$$

Divide both sides by -3.

$$x=-\frac{4}{-3}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{3\ln(5)}+\frac{4}{3}$$

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