document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=609726>Question 609726</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3^n=[square root of]5^n-2. Solve for n.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #383926 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=383926\">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 \ \ \ \ \ \ \ \ \ \ 3^n\ =\ \sqrt{5}^{(n\,-\,2)}\">\r<BR>\n" );
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document.write( "Square both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3^{2n}\ =\ 5^{(n\,-\,2)}\">\r<BR>\n" );
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document.write( "Take the log of both sides (any base, doesn't matter.  If you have a calculator that will calculate logs to any base, or if you can use Excel, you will simplify things greatly if you choose either 3 or 5 as your log base, otherwise you can just use either natural logs or base 10.  I'll proceed using natural logs.)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ln\left(3^{2n}\right)\ =\ \ln\left(5^{(n\,-\,2)}\right)\">\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( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2n\ln\left(3\right)\ =\ (n\ -\ 2)\ln\left(5\right)\">\r<BR>\n" );
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document.write( "A little algebra music, Sammy\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ n\left(2\ln(3)\ -\ \ln(5)\right)\ =\ -2\ln(5)\">\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( "backwards...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ n\left(\ln(9)\ -\ \ln(5)\right)\ =\ -2\ln(5)\">\r<BR>\n" );
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document.write( "The difference of the logs is the log of the quotient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ n\left(\ln\left(\frac{9}{5}\right)\right)\ =\ -2\ln(5)\">\r<BR>\n" );
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document.write( "And finally,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ n\ =\ \frac{-2\ln(5)}{\left(\ln\left(\frac{9}{5}\right)\right)}\">\r<BR>\n" );
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document.write( "You can do the calculator work from here, or just leave it like that for an exact answer.  If you are going to leave it in terms of logs, then go ahead and change the base to 5, using <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \log_b(b)\ =\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ n\ =\ \frac{-2}{\left(\log_5\left(\frac{9}{5}\right)\right)}\">\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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