document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251163>Question 251163</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would I solve the a natural log problem like the one below?<BR>\n" );
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document.write( "((lnX)^2)-6 = -5lnX </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184121 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=184121\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> problem is:\r<BR>\n" );
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document.write( "(ln(x))^2 - 6 = -5*ln(x)\r<BR>\n" );
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document.write( "add 5*ln(x) to both sides of this equation to get:\r<BR>\n" );
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document.write( "(ln(x))^2 + 5*ln(x) - 6 = 0\r<BR>\n" );
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document.write( "let y = ln(x) to get:\r<BR>\n" );
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document.write( "y^2 + 5y - 6 = 0\r<BR>\n" );
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document.write( "factor this equation to get:\r<BR>\n" );
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document.write( "(y+6)*(y-1) = 0\r<BR>\n" );
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document.write( "solve for y to get:\r<BR>\n" );
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document.write( "y = -6 or y = 1\r<BR>\n" );
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document.write( "substitute ln(x) for y to get\r<BR>\n" );
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document.write( "ln(x) = -6 or ln(x) = 1\r<BR>\n" );
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document.write( "by the laws of logarithms, ln(x) = y if and only if e^y = x\r<BR>\n" );
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document.write( "based on that law, we get:\r<BR>\n" );
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document.write( "ln(x) = -6 if and only if e^-6 = x\r<BR>\n" );
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document.write( "and we get:\r<BR>\n" );
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document.write( "ln(x) = 1 if and only if e^1 = x\r<BR>\n" );
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document.write( "we will solve for each separately.\r<BR>\n" );
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document.write( "first we'll solve for e^-6 = x\r<BR>\n" );
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document.write( "using your calculator, e^-6 = .002478752 which makes:\r<BR>\n" );
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document.write( "x = .002478752.\r<BR>\n" );
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document.write( "next we'll solve for e^1 = x\r<BR>\n" );
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document.write( "using your calculator, e^1 = 2.718281828 which makes:\r<BR>\n" );
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document.write( "x = 2.718281828\r<BR>\n" );
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document.write( "x = .002478752 or x = 2.718281828\r<BR>\n" );
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document.write( "we can confirm by substituting in the original equations.\r<BR>\n" );
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document.write( "it is:\r<BR>\n" );
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document.write( "(ln(x))^2 - 6 = -5*ln(x)\r<BR>\n" );
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document.write( "for x = 2.718281828, this equation becomes:\r<BR>\n" );
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document.write( "(ln(2.718281828))^2 - 6 = -5*ln(2.718281828)\r<BR>\n" );
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document.write( "since ln(2.718281828) = 1, this equation becomes:\r<BR>\n" );
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document.write( "1^2 - 6 = -5 which is true, confirming that x = 2.718281828 is good.\r<BR>\n" );
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document.write( "for x = .002478752, this equation becomes:\r<BR>\n" );
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document.write( "(ln(.002478752))^2 - 6 = -5*ln(.002478752)\r<BR>\n" );
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document.write( "since ln(.002478752 = -6, this equation becomes:\r<BR>\n" );
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document.write( "(-6)^2 - 6 = -5*-6 which becomes:\r<BR>\n" );
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document.write( "36 - 6 = 30 which is true, confirming that x = .002478752 is good.\r<BR>\n" );
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document.write( "both answers are good.\r<BR>\n" );
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document.write( "your answer is:\r<BR>\n" );
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document.write( "x = 2.718281828<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "x = .002478752\r<BR>\n" );
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document.write( " </SPAN>\n" );
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