document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=967716>Question 967716</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How long will it take to earn your first million dollars if you annually invest $1000 into an IRA at 10% interest compounded monthly. <BR>\n" );
document.write( "This answer should be calculated using logarithms so that the answer is exact. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #591478 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=591478\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "To calculate this value exactly, a fairly simple formula can be utilized.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=P%281%2B%28r%2Fn%29%29%5E%28n%2At%29 ALIGN=MIDDLE  ALT=\"A=P%281%2B%28r%2Fn%29%29%5E%28n%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" >=principal (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000 ALIGN=MIDDLE  ALT=\"1000\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" >=annual nominal interest rate (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.10 ALIGN=MIDDLE  ALT=\".10\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >=number of compounding during year (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, because of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > months)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >=number of years (unknown)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >=amount after <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > years (first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000000 ALIGN=MIDDLE  ALT=\"1000000\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "This formula yields results when we have the time known, but in this case we need to isolate \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >\" for the time necessary for the principal to turn into $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000000 ALIGN=MIDDLE  ALT=\"1000000\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So we need to rearrange the formula into a fairly complex logarithmic formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28A%29=log%28P%281%2B%28r%2Fn%29%29%5E%28n%2At%29%29 ALIGN=MIDDLE  ALT=\"log%28A%29=log%28P%281%2B%28r%2Fn%29%29%5E%28n%2At%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28A%29=log%28P%29%2Blog%28%281%2B%28r%2Fn%29%29%5E%28n%2At%29%29%29 ALIGN=MIDDLE  ALT=\"log%28A%29=log%28P%29%2Blog%28%281%2B%28r%2Fn%29%29%5E%28n%2At%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28A%29-log%28P%29=log%28%281%2B%28r%2Fn%29%29%5E%28n%2At%29%29%29 ALIGN=MIDDLE  ALT=\"log%28A%29-log%28P%29=log%28%281%2B%28r%2Fn%29%29%5E%28n%2At%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28A%2FP%29=%28n%2At%29log%28%281%2B%28r%2Fn%29%29%29 ALIGN=MIDDLE  ALT=\"log%28A%2FP%29=%28n%2At%29log%28%281%2B%28r%2Fn%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28A%2FP%29%2Flog%28%281%2B%28r%2Fn%29%29%29=%28n%2At%29 ALIGN=MIDDLE  ALT=\"log%28A%2FP%29%2Flog%28%281%2B%28r%2Fn%29%29%29=%28n%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=log%28A%2FP%29%2F%28%28n%2Alog%281%2B%28r%2Fn%29%29%29%29 ALIGN=MIDDLE  ALT=\"t=log%28A%2FP%29%2F%28%28n%2Alog%281%2B%28r%2Fn%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then substitute the numbers into the equation -\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=log%281000000%2F1000%29%2F%28%2812%2Alog%281%2B%280.10%2F12%29%29%29%29 ALIGN=MIDDLE  ALT=\"t=log%281000000%2F1000%29%2F%28%2812%2Alog%281%2B%280.10%2F12%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=log%281000%29%2F%2812%2Alog%281.008333333333333%29%29 ALIGN=MIDDLE  ALT=\"t=log%281000%29%2F%2812%2Alog%281.008333333333333%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=69.36 ALIGN=MIDDLE  ALT=\"t=69.36\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Finally we got the answer - it'll take about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=69 ALIGN=MIDDLE  ALT=\"69\"   DISABLED_style= \"max-width:90%; height:auto;\" > years and a four months!!! \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );