document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=70663>Question 70663</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The demand function for a certain commodity is given by p=80e exponent fraction -q/2 Write q as a function of p </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #50448 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=50448\">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=p+=+80e%5E%28-q%2F2%29 ALIGN=MIDDLE  ALT=\"p+=+80e%5E%28-q%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Solve for q.<BR>\n" );
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document.write( "This is what I understand your problem to be. Assuming this is correct, you can take the ln<BR>\n" );
document.write( "(natural logarithm which has the base e) of both sides and the problem becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ln%28p%29=+ln%2880%2Ae%5E%28-q%2F2%29%29 ALIGN=MIDDLE  ALT=\"+ln%28p%29=+ln%2880%2Ae%5E%28-q%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But the logarithm of a product is equals the sum of the logarithms of the two terms being<BR>\n" );
document.write( "multiplied.  Therefore, we can split the right side into the sum of two logarithms as follows:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ln%28p%29=+ln%2880%29%2B+ln%28e%5E%28-q%2F2%29%29 ALIGN=MIDDLE  ALT=\"+ln%28p%29=+ln%2880%29%2B+ln%28e%5E%28-q%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract ln(80) from both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ln%28p%29+-+ln%2880%29+=++%2B+ln%28e%5E%28-q%2F2%29%29 ALIGN=MIDDLE  ALT=\"+ln%28p%29+-+ln%2880%29+=++%2B+ln%28e%5E%28-q%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By the rules of logarithms, the difference of the logarithms of two quantities can be <BR>\n" );
document.write( "re-written as the logarithm of the quotients of the quantities.  This translates to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ln%28p%29+-+ln%2880%29+=+ln%28p%2F80%29 ALIGN=MIDDLE  ALT=\"+ln%28p%29+-+ln%2880%29+=+ln%28p%2F80%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substituting this as a replacement for the left side results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ln%28p%2F80%29+=+ln%28e%5E%28-q%2F2%29%29 ALIGN=MIDDLE  ALT=\"+ln%28p%2F80%29+=+ln%28e%5E%28-q%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then by a rule of exponents in logarithms, the exponent of a term in a logarithm becomes the<BR>\n" );
document.write( "multiplier of the logarithm of the term on the right side.  In this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-q%2F2%29 ALIGN=MIDDLE  ALT=\"%28-q%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "becomes the multiplier of ln(e) and the right side of the equation is changed as shown below:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ln%28p%2F80%29+=+%28-q%2F2%29%2Aln%28e%29 ALIGN=MIDDLE  ALT=\"+ln%28p%2F80%29+=+%28-q%2F2%29%2Aln%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But ln(e) = 1, and when this substitution is made the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ln%28p%2F80%29+=+%28-q%2F2%29 ALIGN=MIDDLE  ALT=\"+ln%28p%2F80%29+=+%28-q%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply both sides of the equation by -2 and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-2%2Aln%28p%2F80%29+=+q ALIGN=MIDDLE  ALT=\"+-2%2Aln%28p%2F80%29+=+q\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is the answer, but there is one additional constraint. The value of p must be greater <BR>\n" );
document.write( "than zero or else you would be taking the ln of a negative number or zero and those are <BR>\n" );
document.write( "outside of the allowed values of numbers that the ln function can operate on.<BR>\n" );
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document.write( "Hopes this gives you some additional insight about the subject of logarithms and natural<BR>\n" );
document.write( "logarithms in particular. </SPAN>\n" );
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