document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=235481>Question 235481</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4log4 6 - log 4 5, evaluate without a calcuator </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #173457 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=173457\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "The best I can do without a calculator is to express this in terms of a single logarithm.\r<BR>\n" );
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document.write( "Use\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x^n) = n\log_b(x)\">\r<BR>\n" );
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document.write( "To write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4\log_4(6)\ -\ \log_4(5)\ =\ \log_4(6^4)\ -\ \log_4(5)\">\r<BR>\n" );
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document.write( "Then use: <i>The difference of the logs is the log of the quotient</i>, that is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x) - \log_b(y) = log_b\left(\frac{x}{y}\right)\">\r<BR>\n" );
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document.write( "to write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4\log_4(6)\ -\ \log_4(5)\ =\ \log_4(6^4)\ -\ \log_4(5)\ =\ \log_4\left(\frac{6^4}{5}\right)\">\r<BR>\n" );
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document.write( "I need at least a scratch pad to go any further: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 6\times6=36\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 36\times36=1296\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1296\div5=259.2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4\log_4(6)\ -\ \log_4(5)\ =\ \log_4(6^4)\ -\ \log_4(5)\ =\ \log_4\left(\frac{6^4}{5}\right)\ =\ \log_4(259.2)\">\r<BR>\n" );
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document.write( "Now, if you want to do this the old fashioned way with a table of logarithms, you aren't likely to find a table of base 4 logs.  That means you will have to do a base conversion.\r<BR>\n" );
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document.write( "Here's the formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x)\ =\ \frac{\log_c(x)}{\log_c(b)}\">\r<BR>\n" );
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document.write( "So, for your situation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_4(259.2)\ =\ \frac{\log_{10}(259.2)}{\log_{10}(4)}\">\r<BR>\n" );
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document.write( "Then you would have to look up <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log_{10}(2.59)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log_{10}(2.60)\">, then interpolate to find <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log_{10}(2.592)\">.\r<BR>\n" );
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document.write( "To interpolate:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log_{10}(2.59)\ + .2\left(\log_{10}(2.60)\ - \log_{10}(2.59)\right)\">.\r<BR>\n" );
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document.write( "Finally, calculate <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log_{10}(2.592)\ +\ 2\ =\ \log_{10}(259.2)\">.\r<BR>\n" );
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document.write( "Next, look up <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \log_{10}(4)\"> and divide it into the results of the last step -- by hand, of course.\r<BR>\n" );
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document.write( "Good luck.\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" );
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