Solve 4*5^-6x=2000

$$4 \times { 5 }^{ -6x } =2000$$

Solve for x

$$x=-\frac{\log_{5}\left(500\right)}{6}\approx -0.643558853$$

Steps Using Definition of Logarithm

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

$$4\times 5^{-6x}=2000$$

Divide both sides by 4.

$$5^{-6x}=500$$

Take the logarithm of both sides of the equation.

$$\log(5^{-6x})=\log(500)$$

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

$$-6x\log(5)=\log(500)$$

Divide both sides by \log(5).

$$-6x=\frac{\log(500)}{\log(5)}$$

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

$$-6x=\log_{5}\left(500\right)$$

Divide both sides by -6.

$$x=\frac{\log_{5}\left(500\right)}{-6}$$

Solve for x (complex solution)

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

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