Solve {left(3/2right)}^frac{1{2}x}=21/4

$${ \left( \frac{ 3 }{ 2 } \right) }^{ \frac{ 1 }{ 2 } x } =2 \frac{ 1 }{ 4 }$$

Solve for x

$$x=4$$

Steps Using Definition of Logarithm

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

$$\left(\frac{3}{2}\right)^{\frac{1}{2}x}=\frac{9}{4}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{3}{2}\right)^{\frac{1}{2}x})=\log(\frac{9}{4})$$

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

$$\frac{1}{2}x\log(\frac{3}{2})=\log(\frac{9}{4})$$

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

$$\frac{1}{2}x=\frac{\log(\frac{9}{4})}{\log(\frac{3}{2})}$$

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

$$\frac{1}{2}x=\log_{\frac{3}{2}}\left(\frac{9}{4}\right)$$

Multiply both sides by 2.

$$x=\frac{2}{\frac{1}{2}}$$