Question

A car that originally cost $20,000 depreciates (loses value) by 15% each year. In about how many years will the car be worth half its value? Use the equation 10,000 = 20,000(0.85)^x to solve the problem. Round your answer to the nearest dollar.
? years

Answer 1

Is the question - did they mistakenly say "dollar" rather than "years?"

x = # of years

10,000 = 20,000 (0.85)^x, first divide both sides by 20,000
0.5 = (0.85)^x,  next take the ln of both sides
ln 0.5 = ln (0.85)^x,  by rules of logarithms you may move x to make it a coefficient of the logarithm
ln 0.5 = x (ln 0.85),  divide both sides by ln (0.85)
x = (ln 0.5)/ (ln 0.85) = (-0.693)/(-0.1625) = 4.26 years