Solve (2/3)^-2x=9/4

$$( \frac { 2 } { 3 } ) ^ { - 2 x } = \frac { 9 } { 4 }$$

Solve for x

$$x=1$$

Steps Using Definition of Logarithm

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

$$\left(\frac{2}{3}\right)^{-2x}=\frac{9}{4}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{2}{3}\right)^{-2x})=\log(\frac{9}{4})$$

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

$$-2x\log(\frac{2}{3})=\log(\frac{9}{4})$$

Divide both sides by \log(\frac{2}{3}).

$$-2x=\frac{\log(\frac{9}{4})}{\log(\frac{2}{3})}$$

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

$$-2x=\log_{\frac{2}{3}}\left(\frac{9}{4}\right)$$

Divide both sides by -2.

$$x=-\frac{2}{-2}$$

Solve for x (complex solution)

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

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