document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1018108>Question 1018108</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello could you please help me simplify: log little3 3/4+ log little3 4/5+ log little3 5/6+ log little3 6/7+ log little3 7/8+ log little3 8/9 / log 4/5+ log 5/6+ log 6/7+ log 7/8+ log 8/9+ log 9/10 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #634278 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=634278\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\log_3\left(\frac{3}{4}\right)\ +\ \log_3\left(\frac{4}{5}\right)\ +\ \log_3\left(\frac{5}{6}\right)\ +\ \log_3\left(\frac{6}{7}\right)\ +\ \log_3\left(\frac{7}{8}\right)\ +\ \log_3\left(\frac{8}{9}\right)}{\log\left(\frac{4}{5}\right)\ +\ \log\left(\frac{5}{6}\right)\ +\ \log\left(\frac{6}{7}\right)\ +\ \log\left(\frac{7}{8}\right)\ +\ \log\left(\frac{8}{9}\right)\ +\ \log\left(\frac{9}{10}\right)}\">\r<BR>\n" );
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document.write( "Start by using the fact that the log of the quotient is the difference of the logs, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_3\left(\frac{3}{4}\right)\ =\ \log_3(3)\ -\ \log_3(4)\">\r<BR>\n" );
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document.write( "Then if you make all of the appropriate substitutions and collect all like terms (like terms have equal arguments), you end up with:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\log_3(3)\ -\ \log_3(9)}{\log(4)\ -\ \log(10)}\">\r<BR>\n" );
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document.write( "Now there is one little bump in the road at this point.  Some teachers/instructors/professors/mathematicians/textbook authors use the convention that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log\"> without a specified base means base 10, that is:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log(x)\ =\ \log_{10}(x)\">.  Other authorities take <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log\"> to mean the natural or base <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large e\"> logarithm, that is: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log(x)\ =\ \ln(x)\ =\ \log_e(x)\">.  Since the latter interpretation makes your problem rather messy, I'm going to assume the former convention. However, in the future if you want to communicate with someone who is not familiar with the context in which you are asking the question, then you should either specify the base explicitly or specify at the beginning what <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log(x)\"> means in the context of your question.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log_b(b)\ =\ 1\"> for all positive <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log_b\left(x^n\right)\ =\ n\log_b(x)\">, we can simplify further:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\log_3(3)\ -\ \log_3(9)}{\log(4)-\log(10)}\ =\ \frac{1\ -\ 2}{\log(4)\ -\ 1}\ =\ \frac{-1}{\log(4)\ -\ 1}\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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