Solve 2000=5002^8-x

$$2000 = 500 { 2 }^{ 8-x }$$

Solve for x

$$x=6$$

Steps Using Definition of Logarithm

Divide both sides by 500.

$$\frac{2000}{500}=2^{8-x}$$

Divide 2000 by 500 to get 4.

$$4=2^{8-x}$$

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

$$2^{8-x}=4$$

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

$$2^{-x+8}=4$$

Take the logarithm of both sides of the equation.

$$\log(2^{-x+8})=\log(4)$$

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

$$\left(-x+8\right)\log(2)=\log(4)$$

Divide both sides by \log(2).

$$-x+8=\frac{\log(4)}{\log(2)}$$

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

$$-x+8=\log_{2}\left(4\right)$$

Subtract 8 from both sides of the equation.

$$-x=2-8$$

Divide both sides by -1.

$$x=-\frac{6}{-1}$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(2)}+6$$

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