document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=556132>Question 556132</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If interest is compounded continuously, at what annual rate will a principal double in 20 years? Give the answer as a percentage correct to two decimal places. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #362043 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=362043\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The formula for continuously compounded interest is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+C%2Ae%5E%28r%2At%29 ALIGN=MIDDLE  ALT=\"P+=+C%2Ae%5E%28r%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "In which P is the future value of the investment, C is the initial deposit, e is the base of the natural logarithms, r is the annual interest rate expressed as a decimal, and t is the number of years that the money is to be invested.<BR>\n" );
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document.write( "In this case, the initial investment is C and the future value is to be twice C. So we can substitute 2 time C for P and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2AC+=+C%2Ae%5E%28r%2At%29 ALIGN=MIDDLE  ALT=\"2%2AC+=+C%2Ae%5E%28r%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If we then divide both sides of the equation by C, the two C factors disappear and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+=+e%5E%28r%2At%29 ALIGN=MIDDLE  ALT=\"2+=+e%5E%28r%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since the investment is to be for 20 years, we can substitute 20 for t and the equation is then:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+=+e%5E%2820%2Ar%29 ALIGN=MIDDLE  ALT=\"2+=+e%5E%2820%2Ar%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then we take the natural logarithm (base e) of both sides and we have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29+=+ln%28e%5E%2820%2Ar%29%29 ALIGN=MIDDLE  ALT=\"ln%282%29+=+ln%28e%5E%2820%2Ar%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But the rules of logarithms say that an exponent can be brought out as a multiplier of the logarithm. So on the right side we can bring the 20*r out as a multiplier of the logarithm and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29+=+%2820%2Ar%29ln%28e%29 ALIGN=MIDDLE  ALT=\"ln%282%29+=+%2820%2Ar%29ln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But the natural logarithm of e is equal to 1. So we can replace ln(e) by 1 and the equation reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29+=+20%2Ar ALIGN=MIDDLE  ALT=\"ln%282%29+=+20%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve for r by dividing both sides by 20 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28ln%282%29%29%2F20 ALIGN=MIDDLE  ALT=\"r+=+%28ln%282%29%29%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "On a calculator we can find that ln(2) equals 0.69314718. When we substitute that value for ln(2) the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+0.69314718%2F20 ALIGN=MIDDLE  ALT=\"r+=+0.69314718%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Doing this division on a calculator we find that:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+0.034657359 ALIGN=MIDDLE  ALT=\"r+=+0.034657359\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So this is the rate in decimal form that will double a continuously compounded investment in 20 years. You can convert this to percent by moving the decimal two places to the right to get 3.4657359%. Rounded to two decimal places the percentage interest rate would be 3.47%.<BR>\n" );
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document.write( "Hope this helps to clarify how you could do this problem and provides some insight into working with logarithms to solve exponential problems. <BR>\n" );
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