Solve (3/2)^x*(3/2)^2=33/8

$$( \frac { 3 } { 2 } ) ^ { x } \cdot ( \frac { 3 } { 2 } ) ^ { 2 } = 3 \frac { 3 } { 8 }$$

Solve for x

$$x=1$$

Steps Using Definition of Logarithm

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

$$\frac{9}{4}\times \left(\frac{3}{2}\right)^{x}=\frac{27}{8}$$

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

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

Take the logarithm of both sides of the equation.

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

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

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

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

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

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

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

Solve for x (complex solution)

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

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