Solve (9/25)^x=(5/3)^-7

$$( \frac { 9 } { 25 } ) ^ { x } = ( \frac { 5 } { 3 } ) ^ { - 7 }$$

Solve for x

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

Steps Using Definition of Logarithm

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

$$\left(\frac{9}{25}\right)^{x}=\frac{2187}{78125}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{9}{25}\right)^{x})=\log(\frac{2187}{78125})$$

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

$$x\log(\frac{9}{25})=\log(\frac{2187}{78125})$$

Divide both sides by \log(\frac{9}{25}).

$$x=\frac{\log(\frac{2187}{78125})}{\log(\frac{9}{25})}$$

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

$$x=\log_{\frac{9}{25}}\left(\frac{2187}{78125}\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(\frac{3}{5})}+\frac{\log_{\frac{3}{5}}\left(\frac{2187}{78125}\right)}{2}$$

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