document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=541872>Question 541872</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How long will it take for any amount of money to triple if invested at 6% annual interest and compounded quarterly and compounded monthly. I set it up 3=1(1+(.06/4)^4t </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #354330 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=354330\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Ok, that is the correct set up for the 'compounded quarterly' scenario.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(1\ +\ \frac{0.06}{4}\right)^{4t}\ =\ 3\">\r<BR>\n" );
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document.write( "Do the indicated arithmetic and then take the log of both sides (any base, doesn't matter)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ln(1.015)^{4t}\ =\ \ln(3)\">\r<BR>\n" );
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document.write( "Use\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x^n)\ =\ n\log_b(x)\">\r<BR>\n" );
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document.write( "to write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4t\,\cdot\,\ln(1.015)\ =\ \ln(3)\">\r<BR>\n" );
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document.write( "Then a little straight-up algebra:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ \frac{\ln(3)}{4\,\cdot\,\ln(1.015)}\">\r<BR>\n" );
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document.write( "The monthly one works out the same way except that you start with:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(1\ +\ \frac{0.06}{12}\right)^{12t}\ =\ 3\">\r<BR>\n" );
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document.write( "And end up with\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ \frac{\ln(3)}{12\,\cdot\,\ln(1.005)}\">\r<BR>\n" );
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document.write( "Both of them are 18+ years, but the more frequent compounding gets you to your goal about 30 days earlier.\r<BR>\n" );
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document.write( "The reason that the logarithm base doesn't matter is that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\ln(\alpha)}{\ln(\beta)}\ =\ \frac{\log(\alpha)}{\log(\beta)}\ =\ \frac{\log_b(\alpha)}{\log_b(\beta)}\">\r<BR>\n" );
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document.write( "for all real <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \alpha\ >\ 0\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \beta\ >\ 0\"> \r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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