Question 1153381
<pre>
{{{ F = P(1 + r/n)^(nt)  }}}     (1)
where
F = future value
P = present value
r = annual interest rate
n = number of payments per year
t = number of years


This problem asks to find t for F=2P,  r=4% (0.04), and n=4 interest payments per year:
{{{ 2P = P(1 + 0.04/4)^(4t) }}} 
Cancel P from both sides, and simplify where possible:
{{{ 2 = (1.01)^(4t) }}}
Take ln() of both sides:
{{{ ln(2) = 4t*ln(1.01) }}}
Solve for t:
{{{ t = ln(2)/(4*ln(1.01)) = 0.69315/0.03980 = 17.415 }}} years

{{{ highlight( t = 17.415 ) }}} years  (about 17yrs, 151.6days}


--------------
For continuous compounding:
{{{ F = Pe^(rt) }}}   (2)

[ ----------------------

An aside: formula (2) can be derived by taking the limit of (1) as n goes to infinity:
  {{{  F = P(1+r/n)^(nt) }}}   
   Replace r/n by 1/(n/r) and raise the inner expression first to the 'n/r'
     power, and then to the 'rt' power (so the r's would cancel if we carried
     it out):
  {{{  F = P(green(1+(1/(n/r))^(n/r)))^(rt) }}}
   The green expression has a limit of 'e' as n-->infinity:
  {{{ F = Pe^(rt) }}}
--------------------- ]

{{{ 2P = Pe^(0.04t) }}}
{{{ 2 = e^(0.04t) }}}
{{{ ln(2) = 0.04t }}}
{{{ highlight( t = 17.329 ) }}} years (about 17yrs, 120days)