Solve 1388=5002^8-x

$$1388 = 500 { 2 }^{ 8-x }$$

Solve for x

$$x=\log_{2}\left(\frac{125}{347}\right)+8\approx 6.526992432$$

Steps Using Definition of Logarithm

Divide both sides by 500.

$$\frac{1388}{500}=2^{8-x}$$

Reduce the fraction \frac{1388}{500} to lowest terms by extracting and canceling out 4.

$$\frac{347}{125}=2^{8-x}$$

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

$$2^{8-x}=\frac{347}{125}$$

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

$$2^{-x+8}=\frac{347}{125}$$

Take the logarithm of both sides of the equation.

$$\log(2^{-x+8})=\log(\frac{347}{125})$$

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

$$\left(-x+8\right)\log(2)=\log(\frac{347}{125})$$

Divide both sides by \log(2).

$$-x+8=\frac{\log(\frac{347}{125})}{\log(2)}$$

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

$$-x+8=\log_{2}\left(\frac{347}{125}\right)$$

Subtract 8 from both sides of the equation.

$$-x=\frac{\ln(\frac{347}{125})}{\ln(2)}-8$$

Divide both sides by -1.

$$x=\frac{\frac{\ln(\frac{347}{125})}{\ln(2)}-8}{-1}$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(2)}+\log_{2}\left(\frac{125}{347}\right)+8$$

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