document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=108092>Question 108092</A>: <I><SPAN STYLE=\"word-break: break-all;\"> in need of help with:solve algebraically ln 2+ ln(x-6)=ln(x+2) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #78750 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=78750\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29+%2B+ln%28x-6%29+=+ln%28x%2B2%29 ALIGN=MIDDLE  ALT=\"ln%282%29+%2B+ln%28x-6%29+=+ln%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By the rules of logarithms, when you add to logarithms it is equivalent to the logarithm<BR>\n" );
document.write( "of their product. Therefore, the left side of the given equation can be changed as indicated<BR>\n" );
document.write( "below:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%2A%28x-6%29%29+=+ln%28x%2B2%29 ALIGN=MIDDLE  ALT=\"ln%282%2A%28x-6%29%29+=+ln%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Look carefully at this equation. Both sides have the natural logarithm of a quantity.<BR>\n" );
document.write( "So for the equation to be true, the quantities on which the logarithm operates have to <BR>\n" );
document.write( "be equal. This being the case, then we can say:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28x-6%29 ALIGN=MIDDLE  ALT=\"2%2A%28x-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must equal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "so that in the given equation the natural logarithm operator is acting on two equal quantities.<BR>\n" );
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document.write( "Now we can work on solving for x by solving the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28x-6%29+=+x%2B2 ALIGN=MIDDLE  ALT=\"2%2A%28x-6%29+=+x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiplying out the left side changes this equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+-+12+=+x+%2B+2 ALIGN=MIDDLE  ALT=\"2x+-+12+=+x+%2B+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the 12 on the left side by adding 12 to both sides of this equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+=+x+%2B+2+%2B+12 ALIGN=MIDDLE  ALT=\"2x+=+x+%2B+2+%2B+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next get rid of the x on the right side by subtracting x from both sides to reduce the<BR>\n" );
document.write( "equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2+%2B+12+=+14 ALIGN=MIDDLE  ALT=\"x+=+2+%2B+12+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the answer to this problem is x = 14.  <BR>\n" );
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document.write( "You can check this answer by returning to the given problem. Substituting 14 for x in the<BR>\n" );
document.write( "given equation results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29+%2B+ln%2814-6%29+=+ln%2814+%2B+2%29 ALIGN=MIDDLE  ALT=\"ln%282%29+%2B+ln%2814-6%29+=+ln%2814+%2B+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%282%29+%2B+ln%288%29+=+ln%2816%29 ALIGN=MIDDLE  ALT=\"ln%282%29+%2B+ln%288%29+=+ln%2816%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "At this point you can use a calculator to look up each of these three natural logarithms.<BR>\n" );
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document.write( "ln(2) = 0.69314718<BR>\n" );
document.write( "ln(8) = 2.079441542<BR>\n" );
document.write( "ln(16) = 2.772588722<BR>\n" );
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document.write( "Substituting these values for the natural logarithms of the equation results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.69314718%2B2.079441542=2.772588722 ALIGN=MIDDLE  ALT=\"0.69314718%2B2.079441542=2.772588722\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "When you complete the addition on the left side you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.772588722=2.772588722 ALIGN=MIDDLE  ALT=\"2.772588722=2.772588722\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This tells you that when x = 14 the equation is correct.<BR>\n" );
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document.write( "Hope this helps you to understand the problem and the process you can use to find the <BR>\n" );
document.write( "answer for the value of x.<BR>\n" );
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