Solve (5/2)^x=4/25

$$( \frac { 5 } { 2 } ) ^ { x } = \frac { 4 } { 25 }$$

Solve for x

$$x=-2$$

Steps Using Definition of Logarithm

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

$$\left(\frac{5}{2}\right)^{x}=\frac{4}{25}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{5}{2}\right)^{x})=\log(\frac{4}{25})$$

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

$$x\log(\frac{5}{2})=\log(\frac{4}{25})$$

Divide both sides by \log(\frac{5}{2}).

$$x=\frac{\log(\frac{4}{25})}{\log(\frac{5}{2})}$$

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

$$x=\log_{\frac{5}{2}}\left(\frac{4}{25}\right)$$

Solve for x (complex solution)

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

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