document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=718219>Question 718219</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help with this question:\r<BR>\n" );
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document.write( "A simple technique that biologists use to estimate the age of an African elephant is to measure the length of the elephant's footprint and then calculate its age using the equation L = 45 - 257e^-.09a , where L is the length of the elephant's footprint (in centimeters) and a is the age (in years) of the elephant. \r<BR>\n" );
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document.write( "Write an equation and solve to find the ages of the elephants whose footprints are 24 cm long, 28 cm long, 32 cm long, and 36 cm long. Then solve the equation L = 45 = 257e^-.09a for a, and use this equation to find the ages of the elephants whose footprints are given above.\r<BR>\n" );
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document.write( "For the footprints, I did the equations:\r<BR>\n" );
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document.write( "24 cm ; 24 = 45 -257e^-.09a \r<BR>\n" );
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document.write( "28 cm; 28 = 45 -257e^-.09a \r<BR>\n" );
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document.write( "32 cm; 32 = 45 -257e^-.09a \r<BR>\n" );
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document.write( "and 36 cm; 36 = 45 -257e^-.09a \r<BR>\n" );
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document.write( "I am not sure how to solve for a using log/natural log/ e etc. Can you please explain this and the process/ how you got the answer? Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #440824 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=440824\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll do the first one and leave the rest for you to do. The steps for each problem are exactly the same. The only difference is the number that starts on the left side of the equation,<br><BR>\n" );
document.write( "24 cm ; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24+=+45+-257e%5E%28-.09a%29 ALIGN=MIDDLE  ALT=\"24+=+45+-257e%5E%28-.09a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "First we isolate the base and its exponent. Subtracting 45:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-21+=+-257e%5E%28-.09a%29 ALIGN=MIDDLE  ALT=\"-21+=+-257e%5E%28-.09a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing by -257:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-21%2F%28-257%29+=+e%5E%28-.09a%29 ALIGN=MIDDLE  ALT=\"-21%2F%28-257%29+=+e%5E%28-.09a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Converting the fraction to a decimal:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.081712062256809338521400778210117+=+e%5E%28-.09a%29 ALIGN=MIDDLE  ALT=\"0.081712062256809338521400778210117+=+e%5E%28-.09a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(Feel free to round this decimal off as you choose.) Next we use base e, ln, logarithms (because the base of the exponent is e and because our calculators \"know\" ln's).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%280.081712062256809338521400778210117%29+=+ln%28e%5E%28-.09a%29%29 ALIGN=MIDDLE  ALT=\"ln%280.081712062256809338521400778210117%29+=+ln%28e%5E%28-.09a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we use a 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;\" >, which allows us to move the exponent of the argument of a log out in front of the log. (It is this property that is the very reason we use logarithms when the unknown is in an exponent. This property let's us move the exponent (with the variable) out in front where we can \"get at it\" with \"regular\" algebra.) Using ths property we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%280.081712062256809338521400778210117%29+=+%28-.09a%29%2Aln%28e%29 ALIGN=MIDDLE  ALT=\"ln%280.081712062256809338521400778210117%29+=+%28-.09a%29%2Aln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the base of ln is e, the log on the right is just a 1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%280.081712062256809338521400778210117%29+=+-.09a ALIGN=MIDDLE  ALT=\"ln%280.081712062256809338521400778210117%29+=+-.09a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Last we divide by -0.09:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%280.081712062256809338521400778210117%29%2F%28-.09%29+=a ALIGN=MIDDLE  ALT=\"ln%280.081712062256809338521400778210117%29%2F%28-.09%29+=a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using our calculator to find the ln on the left:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2.5045536471717968018511963343956%29%2F%28-.09%29+=a ALIGN=MIDDLE  ALT=\"%28-2.5045536471717968018511963343956%29%2F%28-.09%29+=a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27.828373857464408909457737048839+=+a ALIGN=MIDDLE  ALT=\"27.828373857464408909457737048839+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the elephant is approximately 27.8 years old. </SPAN>\n" );
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