document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=281573>Question 281573</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do you solve <BR>\n" );
document.write( "ln (2x+3)+ln(x-6)-2 ln x =0<BR>\n" );
document.write( "Thank you soooo much<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #204548 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=204548\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Applying \"log rules\":<BR>\n" );
document.write( "ln (2x+3)+ln(x-6)-2 ln x =0<BR>\n" );
document.write( "ln (2x+3)+ln(x-6)- ln x^2 =0<BR>\n" );
document.write( "ln (2x+3)(x-6)- ln x^2 =0<BR>\n" );
document.write( "ln [(2x+3)(x-6)]/x^2 =0<BR>\n" );
document.write( "[(2x+3)(x-6)]/x^2 = e^0<BR>\n" );
document.write( "[(2x+3)(x-6)]/x^2 = 1<BR>\n" );
document.write( "(2x+3)(x-6) = x^2<BR>\n" );
document.write( "2x^2-12x+3x-18 = x^2<BR>\n" );
document.write( "2x^2-9x-18 = x^2<BR>\n" );
document.write( "x^2-9x-18 = 0<BR>\n" );
document.write( "Solve by applying the quadratic equation.  This results in:<BR>\n" );
document.write( "x = {10.685, -1.685}<BR>\n" );
document.write( "But, we can toss out the negative solution since this results in taking a ln of a negative number.  This is an extraneous solution -- toss it out.  Leaving our solution as:<BR>\n" );
document.write( "x = 10.685<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Details of quadratic equation follows:<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-9x%2B-18+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-9x%2B-18+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-9%29%5E2-4%2A1%2A-18=153 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-9%29%5E2-4%2A1%2A-18=153\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=153 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--9%2B-sqrt%28+153+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--9%2B-sqrt%28+153+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-9%29%2Bsqrt%28+153+%29%29%2F2%5C1+=+10.6846584384265 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-9%29%2Bsqrt%28+153+%29%29%2F2%5C1+=+10.6846584384265\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-9%29-sqrt%28+153+%29%29%2F2%5C1+=+-1.68465843842649 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-9%29-sqrt%28+153+%29%29%2F2%5C1+=+-1.68465843842649\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-9x%2B-18 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-9x%2B-18\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-9x%2B-18+=+1%28x-10.6846584384265%29%2A%28x--1.68465843842649%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-9x%2B-18+=+1%28x-10.6846584384265%29%2A%28x--1.68465843842649%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 10.6846584384265, -1.68465843842649.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-9%2Ax%2B-18+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-9%2Ax%2B-18+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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