document.write( "Question 168828: How long does it take $1000 to double if invested at 7% interest compounded continuously? \n" ); document.write( "
Algebra.Com's Answer #124480 by gonzo(654)\"\" \"About 
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formula for continuous compounding is:
\n" ); document.write( "FV = PV * e^(i*x)
\n" ); document.write( "where i is the interest rate
\n" ); document.write( "and x is the number of years.
\n" ); document.write( "-----
\n" ); document.write( "for your problem:
\n" ); document.write( "PV = $1000
\n" ); document.write( "FV = $2000 (double the initial investment of $1000)
\n" ); document.write( "i = .07
\n" ); document.write( "x = what you want to find.
\n" ); document.write( "-----
\n" ); document.write( "FV = PV * e^(i*x)
\n" ); document.write( "becomes:
\n" ); document.write( "2000 = 1000 * e^(.07*x)
\n" ); document.write( "divide both sides of equation by 1000:
\n" ); document.write( "2 = e^(.07*x)
\n" ); document.write( "-----
\n" ); document.write( "logarithms can be used to solve.
\n" ); document.write( "basic rule:
\n" ); document.write( "y = a^x if and only if log.a(y) = x
\n" ); document.write( "where log.a means log to the base a.
\n" ); document.write( "-----
\n" ); document.write( "therefore:
\n" ); document.write( "y = e^x if and only if log.e(y) = x
\n" ); document.write( "log.e = ln
\n" ); document.write( "equation becomes:
\n" ); document.write( "y = e^x if and only if ln(y) = x
\n" ); document.write( "-----
\n" ); document.write( "substituting 2 for y and .07*x for x, we get:
\n" ); document.write( "2 = e^(.07*x) if and only if ln(2) = (.07*x)
\n" ); document.write( "from the calculator, ln(2) = .693147181
\n" ); document.write( "equation becomes:
\n" ); document.write( ".693147181 = .07*x
\n" ); document.write( "x = .693147181/.07 = 9.902102579
\n" ); document.write( "-----
\n" ); document.write( "number of years for money to double with continuous compounding is 9.902102579 years.
\n" ); document.write( "-----
\n" ); document.write( "to prove, plug that value in the original equation.
\n" ); document.write( "2 = e^(.07*x)
\n" ); document.write( "becomes:
\n" ); document.write( "2 = e^(.07*9.902102579) = e^(.693147181) = 2
\n" ); document.write( "-----
\n" ); document.write( "formula checks out ok.
\n" ); document.write( "you can use your calculator to prove.
\n" ); document.write( "you can also use online continuous compounding calculator found at:
\n" ); document.write( "http://www.moneychimp.com/articles/finworks/continuous_compounding.htm \n" ); document.write( "
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