Question 169820
Future Value (FV) in terms of Present Value (PV), interest rate (i), and number of years (Y) under continuous compounding is:


{{{FV=PVe^(iY)}}} where e is the base of the natural logarithms defined by:


If {{{y = e^x}}} then {{{ln(x) = y}}}


Given:

FV = 31000
PV = 8900
i = 11.9% = .119


then


{{{31000=8900e^(.119Y)}}}


Taking the natural log of both sides:


{{{ln(31000)=ln(8900e^(.119Y))}}}


Applying the rules of logarithms:


{{{ln(31000)=ln(8900)+ln(e^(.119Y))}}}


{{{ln(31000)=ln(8900)+0.119Yln(e)}}}


But {{{ln(e)=1}}} because {{{ln(e^x)=x}}} so {{{ln(e^1)=1}}}


Rearranging:


{{{0.119Y=ln(31000) - ln(8900)}}}
{{{Y=(ln(31000)-ln(8900))/0.119}}}


A little calculator work gets you to about 10.49 years rounded to the nearest hundreth.