document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=658621>Question 658621</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For this problem in Newtons law of cooling, what does the e mean?  Why do we use ln and what is ln?\r<BR>\n" );
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document.write( "T (t) = Te + (T0 &#8722; Te ) e^-kt\r<BR>\n" );
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document.write( "160= 69 + (190-69)e^k2<BR>\n" );
document.write( "160-69 = 69-69(121)e^k2<BR>\n" );
document.write( "91 = (121)ek2<BR>\n" );
document.write( "91/121 = (121/121)e^2k<BR>\n" );
document.write( "e^2k = 91/121<BR>\n" );
document.write( "2k = ln(91/121)<BR>\n" );
document.write( "k = .5(ln(91/121))<BR>\n" );
document.write( "k = .5(-.284931)<BR>\n" );
document.write( "k = -.1425\r<BR>\n" );
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document.write( "69 + (180 - 69)e^-.1425t = 130<BR>\n" );
document.write( "69-69 + 121e-.1425t = 130-69<BR>\n" );
document.write( "121e.-1425t = 61<BR>\n" );
document.write( "e^-.1425t = 61/121<BR>\n" );
document.write( "-.1425t = ln(61/121)<BR>\n" );
document.write( "t = (-.684917)/(-.1425)<BR>\n" );
document.write( "t = 4.8 minutes  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #410354 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=410354\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For this problem in Newtons law of cooling, what does the e mean? Why do we use ln and what is ln? <BR>\n" );
document.write( "T (t) = Te + (T0 &#8722; Te ) e^-kt<BR>\n" );
document.write( "T(t) is he temperature of the object at time \"t\".<BR>\n" );
document.write( "Te is the temperature of the environment.<BR>\n" );
document.write( "To is the temperature of the object at some initial time (e.g. time of death)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "The e in e^kt is the irrational number e = 2.718281828...<BR>\n" );
document.write( "------------------------------<BR>\n" );
document.write( "160= 69 + (190-69)e^k2<BR>\n" );
document.write( "160-69 = 69-69(121)e^k2<BR>\n" );
document.write( "91 = (121)ek2<BR>\n" );
document.write( "91/121 = (121/121)e^2k<BR>\n" );
document.write( "e^2k = 91/121<BR>\n" );
document.write( "2k = ln(91/121)<BR>\n" );
document.write( "---<BR>\n" );
document.write( "ln is the \"natural log\"; it is the power of \"e\" that gives you some <BR>\n" );
document.write( "particular number.<BR>\n" );
document.write( "For example:<BR>\n" );
document.write( "ln(e^k2) = k2<BR>\n" );
document.write( "ln(5) = 1.609.. because e^(1.609..) = 5<BR>\n" );
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document.write( "k = .5(ln(91/121))<BR>\n" );
document.write( "k = .5(-.284931)<BR>\n" );
document.write( "k = -.1425 <BR>\n" );
document.write( "69 + (180 - 69)e^-.1425t = 130<BR>\n" );
document.write( "69-69 + 121e-.1425t = 130-69<BR>\n" );
document.write( "121e.-1425t = 61<BR>\n" );
document.write( "e^-.1425t = 61/121<BR>\n" );
document.write( "-.1425t = ln(61/121)<BR>\n" );
document.write( "t = (-.684917)/(-.1425)<BR>\n" );
document.write( "t = 4.8 minutes <BR>\n" );
document.write( "===================<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H. </SPAN>\n" );
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