Solve 25{left(4/3right)}^-x=12000

$$25 { \left( \frac{ 4 }{ 3 } \right) }^{ -x } =12000$$

Solve for x

$$x=\log_{\frac{3}{4}}\left(480\right)\approx -21.460447818$$

Steps Using Definition of Logarithm

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

$$25\times \left(\frac{4}{3}\right)^{-x}=12000$$

Divide both sides by 25.

$$\left(\frac{4}{3}\right)^{-x}=480$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{4}{3}\right)^{-x})=\log(480)$$

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

$$-x\log(\frac{4}{3})=\log(480)$$

Divide both sides by \log(\frac{4}{3}).

$$-x=\frac{\log(480)}{\log(\frac{4}{3})}$$

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

$$-x=\log_{\frac{4}{3}}\left(480\right)$$

Divide both sides by -1.

$$x=\frac{\ln(480)}{-\ln(\frac{4}{3})}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(\frac{3}{4})}+\log_{\frac{3}{4}}\left(480\right)$$

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