document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=631098>Question 631098</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have tried and tried on this problem and just cant get it right. I even had a tutor online help me and they got it wrong too... this is the problem ...\r<BR>\n" );
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document.write( "Solve the system of equations. Round approximate values to the nearest ten-thousandth. \r<BR>\n" );
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document.write( "y = log2x<BR>\n" );
document.write( "y = x&#8722;5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #397443 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=397443\">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( "There is no algebraic solution to this equation.  You have to use numerical methods.\r<BR>\n" );
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document.write( "I used a graphing program and then ran some numbers through the function in Excel.  I was able to come up with two intervals: [0.000005,0.000006] and [6,6.1] where the function changes sign from one end of the interval to the other.  The Intermediate value theorem says there has to be a root on each of those intervals.\r<BR>\n" );
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document.write( "Then I used the Newton-Raphson method:\r<BR>\n" );
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document.write( "I first defined the function as\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ \log(2x)\ -\ x\ +\ 5\">\r<BR>\n" );
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document.write( "Then Newton-Raphson says \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_1\ =\ x_0\ -\ \frac{f(x_0)}{f'(x_0)}\">\r<BR>\n" );
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document.write( "and then a closer approximation can be obtained from:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_2\ =\ x_1\ -\ \frac{f(x_1)}{f'(x_1)}\">\r<BR>\n" );
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document.write( "And you can repeat the process as often as you like to get the desired precision.\r<BR>\n" );
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document.write( "The derivative of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \log(2x)\ -\ x\ +\ 5\"> is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f'(x)\ =\ \frac{\log(e)}{x}\ -\ 1\">\r<BR>\n" );
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document.write( "You can run a calculator as well as I can.  Enjoy.\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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