Solve {left(2/3right)}^-2{left(3/2right)}^2x={left(4/9right)}^2

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

Solve for x

$$x=-3$$

Steps Using Definition of Logarithm

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

$$\frac{9}{4}\times \left(\frac{3}{2}\right)^{2x}=\frac{16}{81}$$

Divide both sides of the equation by \frac{9}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\left(\frac{3}{2}\right)^{2x}=\frac{64}{729}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{3}{2}\right)^{2x})=\log(\frac{64}{729})$$

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

$$2x\log(\frac{3}{2})=\log(\frac{64}{729})$$

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

$$2x=\frac{\log(\frac{64}{729})}{\log(\frac{3}{2})}$$

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

$$2x=\log_{\frac{3}{2}}\left(\frac{64}{729}\right)$$

Divide both sides by 2.

$$x=-\frac{6}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(\frac{3}{2})}+\log_{\frac{3}{2}}\left(\frac{8}{27}\right)$$

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