Solve (45*32){left(8/3right)}^2x=23

$$(45 \times 32) { \left( \frac{ 8 }{ 3 } \right) }^{ 2x } =23$$

Solve for x

$$x=\frac{\log_{\frac{8}{3}}\left(\frac{23}{1440}\right)}{2}\approx -2.108880911$$

Steps Using Definition of Logarithm

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

$$1440\times \left(\frac{8}{3}\right)^{2x}=23$$

Divide both sides by 1440.

$$\left(\frac{8}{3}\right)^{2x}=\frac{23}{1440}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{8}{3}\right)^{2x})=\log(\frac{23}{1440})$$

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

$$2x\log(\frac{8}{3})=\log(\frac{23}{1440})$$

Divide both sides by \log(\frac{8}{3}).

$$2x=\frac{\log(\frac{23}{1440})}{\log(\frac{8}{3})}$$

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

$$2x=\log_{\frac{8}{3}}\left(\frac{23}{1440}\right)$$

Divide both sides by 2.

$$x=\frac{\ln(\frac{23}{1440})}{2\ln(\frac{8}{3})}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(\frac{8}{3})}+\frac{\log_{\frac{8}{3}}\left(\frac{23}{1440}\right)}{2}$$

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