Solve (2/3)^x=(3/2)^5

$$( \frac { 2 } { 3 } ) ^ { x } = ( \frac { 3 } { 2 } ) ^ { 5 }$$

Solve for x

$$x=-5$$

Steps Using Definition of Logarithm

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

$$\left(\frac{2}{3}\right)^{x}=\frac{243}{32}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{2}{3}\right)^{x})=\log(\frac{243}{32})$$

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

$$x\log(\frac{2}{3})=\log(\frac{243}{32})$$

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

$$x=\frac{\log(\frac{243}{32})}{\log(\frac{2}{3})}$$

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

$$x=\log_{\frac{2}{3}}\left(\frac{243}{32}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(\frac{2}{3})}+\log_{\frac{2}{3}}\left(\frac{243}{32}\right)$$

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