Problem:
Suppose that at the end of any year, a unit of money has lost 10% of the value it had at the beginning of that year. Find the smallest integer n such that after n years the unit of money will have lost at least 90% of its value. (To the nearest thousandth log10​3 is .477.)
Answer Choices:
A. 14
B. 16
C. 18
D. 20
E. 22
Solution:
If A denotes the value of the unit of money at a given time, then .9A denotes its value a year later and (.9)nA denotes its value n years later. We seek the smallest integer n such that n satisfies these equivalent inequalities:
(.9)nA(109​)nlog10​(109​)nn(2log10​3−1)n​⩽.1A⩽101​⩽log10​101​⩽−1⩾1−2log10​31​≈21.7​