Solve 5={left(45/20right)}^xdiv2

$$5= { \left( \frac{ 45 }{ 20 } \right) }^{ x \div 2 }$$

Solve for x

$$x=\log_{\frac{3}{2}}\left(5\right)\approx 3.969362296$$

Steps Using Definition of Logarithm

Reduce the fraction \frac{45}{20} to lowest terms by extracting and canceling out 5.

$$5=\left(\frac{9}{4}\right)^{\frac{x}{2}}$$

Swap sides so that all variable terms are on the left hand side.

$$\left(\frac{9}{4}\right)^{\frac{x}{2}}=5$$

Subtract 5 from both sides.

$$\left(\frac{9}{4}\right)^{\frac{x}{2}}-5=0$$

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

$$\left(\frac{9}{4}\right)^{\frac{1}{2}x}-5=0$$

Add 5 to both sides of the equation.

$$\left(\frac{9}{4}\right)^{\frac{1}{2}x}=5$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{9}{4}\right)^{\frac{1}{2}x})=\log(5)$$

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

$$\frac{1}{2}x\log(\frac{9}{4})=\log(5)$$

Divide both sides by \log(\frac{9}{4}).

$$\frac{1}{2}x=\frac{\log(5)}{\log(\frac{9}{4})}$$

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

$$\frac{1}{2}x=\log_{\frac{9}{4}}\left(5\right)$$

Multiply both sides by 2.

$$x=\frac{\ln(5)}{\frac{1}{2}\ln(\frac{9}{4})}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(\frac{3}{2})}+\log_{\frac{3}{2}}\left(5\right)$$

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