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

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

Solve for x

$$x=-1$$

Steps Using Definition of Logarithm

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

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

Take the logarithm of both sides of the equation.

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

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

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

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

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

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

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

Divide both sides by 2.

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

Solve for x (complex solution)

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

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