document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=628010>Question 628010</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use algebraic procedures to find the exact solution(s) of the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)<BR>\n" );
document.write( " 4^(4x &#8722; 3) = 5^(&#8722;x &#8722; 1)\r<BR>\n" );
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document.write( "x=(?,?) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #395366 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=395366\">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 \ \ \ \ \ \ \ \ \ \ 4^{(4x\ -\ 3)}\ =\ 5^{(-x\ -\ 1)}\">\r<BR>\n" );
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document.write( "Take the log of both sides (any base, doesn't matter)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ln\left(4^{(4x\ -\ 3)}\right)\ =\ \ln\left(5^{(-x\ -\ 1)}\right)\">\r<BR>\n" );
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document.write( "Use <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \log_b(x^n)\ =\ n\log_b(x)\"> on both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (4x\ -\ 3)\ln\left(4\right)\ =\ (-x\ -\ 1)\ln\left(5\right)\">\r<BR>\n" );
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document.write( "Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{\ln(4)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (4x\ -\ 3)\ =\ (-x\ -\ 1)\frac{\ln\left(5\right)}{\ln\left(4\right)}\">\r<BR>\n" );
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document.write( "just for the sake of neatness, let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\ =\ \frac{\ln\left(5\right)}{\ln\left(4\right)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (4x\ -\ 3)\ =\ (-x\ -\ 1)k\">\r<BR>\n" );
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document.write( "Then it is just plain old algebra from here:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4x\ +\ kx\ =\ 3\ -\ k\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{3\ -\ k}{4\ +\ k}\ =\ \frac{3\ -\ \frac{\ln\left(5\right)}{\ln\left(4\right)}}{4\ +\ \frac{\ln\left(5\right)}{\ln\left(4\right)}}\">\r<BR>\n" );
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document.write( "Since you want the exact answer, and you don't care about actually being able to approximate the log values, you can get a much neater answer using base 4 logs:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{3\ -\ \log_4(5)}{4\ +\ \log_4(5)}\">\r<BR>\n" );
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document.write( "I'll let you figure out why these two answers are the same.  Hint:  What is the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \log_b(b)\">?\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<BR>\n" );
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