Solve 10000left(1.20right)^-x=3000

$$10000 { \left(1.20 \right) }^{ -x } =3000$$

Solve for x

$$x=\log_{\frac{5}{6}}\left(0.3\right)\approx 6.603568034$$

Steps Using Definition of Logarithm

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

$$10000\times 1.2^{-x}=3000$$

Divide both sides by 10000.

$$1.2^{-x}=\frac{3}{10}$$

Take the logarithm of both sides of the equation.

$$\log(1.2^{-x})=\log(\frac{3}{10})$$

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

$$-x\log(1.2)=\log(\frac{3}{10})$$

Divide both sides by \log(1.2).

$$-x=\frac{\log(\frac{3}{10})}{\log(1.2)}$$

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

$$-x=\log_{1.2}\left(\frac{3}{10}\right)$$

Divide both sides by -1.

$$x=\frac{\ln(\frac{3}{10})}{-\ln(\frac{6}{5})}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(\frac{5}{6})}+\log_{\frac{5}{6}}\left(0.3\right)$$

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