document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=242151>Question 242151</A>: <I><SPAN STYLE=\"word-break: break-all;\"> carl plans to invest $500 at 8.25% INTREST, COMPOUNDED CONTINUOUSLY. how long will it take for his money to triple? use y= Pe^rt. show your work </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #177470 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=177470\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> After tripling $500 we will have $1500. So the equation we have to solve is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1500+=+500e%5E%280.0825t%29 ALIGN=MIDDLE  ALT=\"1500+=+500e%5E%280.0825t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "First we'll divide both sides by 500:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+=+e%5E%280.0825t%29 ALIGN=MIDDLE  ALT=\"3+=+e%5E%280.0825t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now, since the variable is up in the exponent, we will use logarithms to solve for it. The natural choice (excuse the pun) is to use natural logarithms (aka ln) because e is the base:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%283%29+=+ln%28e%5E%280.0825t%29%29 ALIGN=MIDDLE  ALT=\"ln%283%29+=+ln%28e%5E%280.0825t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the property of logarithms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can move the exponent in the argument out in front as a coefficient:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%283%29+=+%280.0825t%29ln%28e%29 ALIGN=MIDDLE  ALT=\"ln%283%29+=+%280.0825t%29ln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%29+=+1 ALIGN=MIDDLE  ALT=\"ln%28e%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > by definition we now have:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%283%29+=+%280.0825t%29 ALIGN=MIDDLE  ALT=\"ln%283%29+=+%280.0825t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide both sides by 0.0825:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%283%29%2F0.0825+=+t ALIGN=MIDDLE  ALT=\"ln%283%29%2F0.0825+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is an exact answer. You can use your calculator to find a decimal approximation for t (which is the time it will take to triple your money).<br><BR>\n" );
document.write( "If your calculator doesn't \"do\" ln, we can use base 10 logs instead of ln:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%283%29%29+=+log%28%28e%5E%280.0825t%29%29%29 ALIGN=MIDDLE  ALT=\"log%28%283%29%29+=+log%28%28e%5E%280.0825t%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%283%29%29+=+%280.0825t%29log%28%28e%29%29 ALIGN=MIDDLE  ALT=\"log%28%283%29%29+=+%280.0825t%29log%28%28e%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since we used base 10 logs, log(e) will not \"disappear\" like ln(e). So we are \"stuck\" with it. Divide both sides by 0.0825log(e):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%283%29%29%2F%280.0825%2Alog%28%28e%29%29%29+=+t ALIGN=MIDDLE  ALT=\"log%28%283%29%29%2F%280.0825%2Alog%28%28e%29%29%29+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" > (e is approximately equal to 2.7182818284590451. Round this off as you choose and then use your calculator to find t.<br><BR>\n" );
document.write( "Although <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%283%29%29%2F%280.0825%2Alog%28%28e%29%29%29 ALIGN=MIDDLE  ALT=\"log%28%283%29%29%2F%280.0825%2Alog%28%28e%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%283%29%2F0.0825 ALIGN=MIDDLE  ALT=\"ln%283%29%2F0.0825\"   DISABLED_style= \"max-width:90%; height:auto;\" > do not look the same they are actually equal. \r<BR>\n" );
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