document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=718152>Question 718152</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve for A and B in this exponential decay model y= ae^-bx. I have the following points: when x=0.6, y=1000 and when x=1.8, y=1.\r<BR>\n" );
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document.write( "I have done the following but I am not convienced it is the right path...\r<BR>\n" );
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document.write( "y=ae-bx therefor e-bx=y/a ... take log e of both sides ... loge e-bx = loge y/a\r<BR>\n" );
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document.write( "but logee=1 therefor -bx=loge y/a ... move the x over to get -b=x loge y/a\r<BR>\n" );
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document.write( "then enter values for x and y from example 1... -b= (0.6) loge(1000/a)\r<BR>\n" );
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document.write( "Is this the correct path to take for solving this?? Any advice would be greafull appreciated. Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #440845 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=440845\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a%2Ae%5E%28-bx%29 ALIGN=MIDDLE  ALT=\"y+=+a%2Ae%5E%28-bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > when x=0.6, y=1000 and when x=1.8, y=1<br><BR>\n" );
document.write( "No matter what you do, you are going to have to solve a system of two equations with two unknowns, a and b. Although what you've done is not an invalid path, I'm not convinced that it makes the problem any easier.<br><BR>\n" );
document.write( "We'll start by correcting an error in your work and then finishing the problem. Later we'll look at ways you might find as good or better than what you've been trying.<br><BR>\n" );
document.write( "Your work is find up to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-bx+=+log%28e%2C+%28y%2Fa%29%29 ALIGN=MIDDLE  ALT=\"-bx+=+log%28e%2C+%28y%2Fa%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "But to \"move x over\" you have to divide by x, not multiply it:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b+=+log%28e%2C+%28y%2Fa%29%29%2Fx ALIGN=MIDDLE  ALT=\"-b+=+log%28e%2C+%28y%2Fa%29%29%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting in the values for x and y we get two equations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b+=+log%28e%2C+%281000%2Fa%29%29%2F0.6 ALIGN=MIDDLE  ALT=\"-b+=+log%28e%2C+%281000%2Fa%29%29%2F0.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b+=+log%28e%2C+%281%2Fa%29%29%2F1.8 ALIGN=MIDDLE  ALT=\"-b+=+log%28e%2C+%281%2Fa%29%29%2F1.8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the Substitution Method on this system, we would substitute one equation's expression for -b in for the other equation's -b:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28e%2C+%281000%2Fa%29%29%2F0.6++=+log%28e%2C+%281%2Fa%29%29%2F1.8 ALIGN=MIDDLE  ALT=\"log%28e%2C+%281000%2Fa%29%29%2F0.6++=+log%28e%2C+%281%2Fa%29%29%2F1.8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then we would solve this for a. Then we would use this value for a and one of the earlier equations (which has a and b) to solve for b.<br><BR>\n" );
document.write( "Although the path you were trying to take eventually would have worked (if there was no error), here's a couple of alternatives you might find easier and/or faster. They both start with substituting in the x's and y's into the original equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a%2Ae%5E%28-bx%29 ALIGN=MIDDLE  ALT=\"y+=+a%2Ae%5E%28-bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "giving us:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000+=+a%2Ae%5E%28-b%280.6%29%29 ALIGN=MIDDLE  ALT=\"1000+=+a%2Ae%5E%28-b%280.6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+a%2Ae%5E%28-b%281.8%29%29 ALIGN=MIDDLE  ALT=\"1+=+a%2Ae%5E%28-b%281.8%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplify to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000+=+a%2Ae%5E%28-0.6b%29 ALIGN=MIDDLE  ALT=\"1000+=+a%2Ae%5E%28-0.6b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+a%2Ae%5E%28-1.8b%29 ALIGN=MIDDLE  ALT=\"1+=+a%2Ae%5E%28-1.8b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From here we could...<ul><li>Solve one equation for \"a\" and use the Substitution Method to solve the system:<BR>\n" );
document.write( "Dividing the first equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-0.6b%29 ALIGN=MIDDLE  ALT=\"e%5E%28-0.6b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000%2Fe%5E%28-0.6b%29+=+a ALIGN=MIDDLE  ALT=\"1000%2Fe%5E%28-0.6b%29+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000e%5E%280.6b%29+=+a ALIGN=MIDDLE  ALT=\"1000e%5E%280.6b%29+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then substitute this into the other equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+%281000e%5E%280.6b%29%29e%5E%28-1.8b%29 ALIGN=MIDDLE  ALT=\"1+=+%281000e%5E%280.6b%29%29e%5E%28-1.8b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which, after adding e's exponents, simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+1000e%5E%28-1.2b%29 ALIGN=MIDDLE  ALT=\"1+=+1000e%5E%28-1.2b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide by 1000:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.001+=+e%5E%28-1.2b%29 ALIGN=MIDDLE  ALT=\"0.001+=+e%5E%28-1.2b%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%28e%2C+%280.001%29%29+=+log%28e%2C+%28e%5E%28-1.2b%29%29%29 ALIGN=MIDDLE  ALT=\"log%28e%2C+%280.001%29%29+=+log%28e%2C+%28e%5E%28-1.2b%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%28e%2C+%280.001%29%29+=+%28-1.2b%29%2Alog%28e%2C+%28e%29%29 ALIGN=MIDDLE  ALT=\"log%28e%2C+%280.001%29%29+=+%28-1.2b%29%2Alog%28e%2C+%28e%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%28e%2C+%280.001%29%29+=+-1.2b ALIGN=MIDDLE  ALT=\"log%28e%2C+%280.001%29%29+=+-1.2b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28e%2C+%280.001%29%29%2F%28-1.2%29+=+b ALIGN=MIDDLE  ALT=\"log%28e%2C+%280.001%29%29%2F%28-1.2%29+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "etc.</li><li>Perhaps even faster...<BR>\n" );
document.write( "Starting from:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000+=+a%2Ae%5E%28-0.6b%29 ALIGN=MIDDLE  ALT=\"1000+=+a%2Ae%5E%28-0.6b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+a%2Ae%5E%28-1.8b%29 ALIGN=MIDDLE  ALT=\"1+=+a%2Ae%5E%28-1.8b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide the two equations!<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000%2F1+=+%28a%2Ae%5E%28-0.6b%29%29%2F%28a%2Ae%5E%28-1.8b%29%29 ALIGN=MIDDLE  ALT=\"1000%2F1+=+%28a%2Ae%5E%28-0.6b%29%29%2F%28a%2Ae%5E%28-1.8b%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "On the right side the a's cancel and, after subtracting e's exponents, we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000+=+e%5E%281.2b%29 ALIGN=MIDDLE  ALT=\"1000+=+e%5E%281.2b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "etc.</li></ul>I'll let you choose which solution, your corrected path or one of the two alternatives I've shown, to use to finish the problem. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );