document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=895438>Question 895438</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am stuck on this problem. 2 ln(x)= ln(x-3)+ln(x+1)<BR>\n" );
document.write( "I tried doing 2 ln(x)= ln(x^2-2x+3)<BR>\n" );
document.write( "then I tried dividing the x to move the xs to the right<BR>\n" );
document.write( "2 ln=ln(x-2+3).<BR>\n" );
document.write( "Please help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #542820 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=542820\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> interesting problem because you can get a solution but not if you have to go back to the original equation to confirm that it is true.\r<BR>\n" );
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document.write( "in other words, the solution is valid for an intermediate equation but not for the original equation.\r<BR>\n" );
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document.write( "your original equation is 2*ln(x) = ln(x-3) + ln(x+1)\r<BR>\n" );
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document.write( "since 2*ln(x) is equivalent to ln(x^2), your equation becomes:\r<BR>\n" );
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document.write( "ln(x^2) = ln(x-3) + ln(x+1)\r<BR>\n" );
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document.write( "since ln(x-3) + ln(x+1) is the same as ln((x-3)*(x+1)), your equation becomes:\r<BR>\n" );
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document.write( "ln(x^2) = ln(x^2-2x-3)\r<BR>\n" );
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document.write( "this is true if and only if x^2 = x^2-2x-3\r<BR>\n" );
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document.write( "x^2 = x^2-2x-3 when x = -3/2.\r<BR>\n" );
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document.write( "here's where the conflict comes into play.\r<BR>\n" );
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document.write( "if you replace x with -3/2 in the equation of ln(x^2) = ln(x^2-2x-3), you will get .8109... = .8109... which conforms that the value of x = -3/2 is a good solution.\r<BR>\n" );
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document.write( "unfortunately, you have to go back to the original equation to confirm that the solution is good.\r<BR>\n" );
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document.write( "if you replace x with -3/2 in the equation of 2*ln(x) = ln(x-3) + ln(x+1), you will get an error indication telling you that the solution is not real.\r<BR>\n" );
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document.write( "if you graph the equation of ln(x^2) and you graph the equation of ln(x^2-2x-3), you will find that those graphs will intersect at x = -3/2.\r<BR>\n" );
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document.write( "that graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://theo.x10hosting.com/2014/082707.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "if ou graph the equation of 2*ln(x) and you graph the equation of ln(x-3) + ln(x+1), you will find that those graphs will not intersect because the value of x = -3/2 is not part of the domain since you can't take the log of a negative number.\r<BR>\n" );
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document.write( "that graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://theo.x10hosting.com/2014/082708.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "note that, for values of x > 0, the graph of y = 2*ln(x) is identical to the graph of y = ln(x^2)\r<BR>\n" );
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document.write( "not also that, for value of x > 0, the graph of y = ln(x-3) + ln(x+1) is identical to the graph of y = ln(x^2-2x-3).\r<BR>\n" );
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document.write( "these graphs are identical, but only where the valid domain for each coincides.\r<BR>\n" );
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document.write( "their domains are both valid when x > 0.\r<BR>\n" );
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document.write( "for example, taking a random value of x > 3, we get:\r<BR>\n" );
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document.write( "when x = 5:\r<BR>\n" );
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document.write( "y = 2*ln(x) = 3.218875...<BR>\n" );
document.write( "y = ln(x^2) = 3.218875...\r<BR>\n" );
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document.write( "y = ln(x-3) + ln(x+1) = 2.484906...<BR>\n" );
document.write( "y = ln(x^2-2x-3) = 2.484906...\r<BR>\n" );
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document.write( "this has also been confirmed graphically for all values of x > 3.\r<BR>\n" );
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document.write( "the equatons of y = 2*ln(x) and y = ln(x^2) are identical for all values of x > 0.\r<BR>\n" );
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document.write( "when x is less than or equal to 0, the equation of 2*ln(x) is invalid.\r<BR>\n" );
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document.write( "the equations of y = ln(x-3) + ln(x+1) and y = ln(x^2 -2x - 3) are identical for all values of x > 3\r<BR>\n" );
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document.write( "when x is less than or equal to 3, the equation of ln(x-3) + ln(x+1) is invalid because x-3 is less than or equal to 0.\r<BR>\n" );
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document.write( "ln(x+1) is valid for all values of x > -1, but that doesn't count because both halves of that equation have to be valid and ln(x-3) is invalid for any values of x less than or equal to 3.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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