document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=113219>Question 113219</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you solve this exactly?\r<BR>\n" );
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document.write( "log(x+20)-log(x+2)=log x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #82396 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=82396\">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=log%28x%2B20%29-log%28x%2B2%29=log%28x%29 ALIGN=MIDDLE  ALT=\"log%28x%2B20%29-log%28x%2B2%29=log%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By the rules of logarithms, the difference of two logs is equal to the log of their quotient.<BR>\n" );
document.write( "In other words:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28A%29+-+log%28B%29+=+log%28A%2FB%29 ALIGN=MIDDLE  ALT=\"log%28A%29+-+log%28B%29+=+log%28A%2FB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By applying this rule to the left side of the given problem, you convert the problem to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28x%2B20%29%2F%28x%2B2%29%29+=+log%28x%29 ALIGN=MIDDLE  ALT=\"log%28%28x%2B20%29%2F%28x%2B2%29%29+=+log%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that this tells you that the log of a quantity on the left side equals the log of a<BR>\n" );
document.write( "quantity on the right side. For this to be true, the two quantities must be equal because<BR>\n" );
document.write( "their logs are equal. Therefore, you can say that:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B20%29%2F%28x%2B2%29+=+x ALIGN=MIDDLE  ALT=\"%28x%2B20%29%2F%28x%2B2%29+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the denominator on the left side by multiplying both sides by (x + 2) to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28x%2B2%29%2A%28x+%2B+20%29%2Fcross%28x%2B2%29+=+%28x%2B2%29%2Ax ALIGN=MIDDLE  ALT=\"cross%28x%2B2%29%2A%28x+%2B+20%29%2Fcross%28x%2B2%29+=+%28x%2B2%29%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply out the right side and you are left with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+20+=+x%5E2+%2B+2x ALIGN=MIDDLE  ALT=\"x+%2B+20+=+x%5E2+%2B+2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract x + 20 from both sides and you get a quadratic equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+=+x%5E2+%2B+2x++-+x+-20 ALIGN=MIDDLE  ALT=\"+0+=+x%5E2+%2B+2x++-+x+-20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "combine the + 2x and the -x and you are left with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+x%5E2+%2Bx+-+20 ALIGN=MIDDLE  ALT=\"0+=+x%5E2+%2Bx+-+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Transpose (switch sides) this equation to get the more conventional quadratic form of:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+x+-+20+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+x+-+20+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve this equation by factoring the left side:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+5%29%28x-4%29+=+0 ALIGN=MIDDLE  ALT=\"%28x+%2B+5%29%28x-4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This equation will be true if either or both of the factors on the left side is zero because<BR>\n" );
document.write( "a multiplication by zero on the left side makes the entire left side zero and therefore<BR>\n" );
document.write( "equal to the right side.<BR>\n" );
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document.write( "What value of x makes a factor equal to zero? Set the factor (x + 5) equal to zero and<BR>\n" );
document.write( "when you solve the equation you get x = -5. Then set the factor (x - 4) equal to zero and<BR>\n" );
document.write( "solve to get x = +4. So you have two possible answers to this problem ...  x = -5 and <BR>\n" );
document.write( "x = +4.<BR>\n" );
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document.write( "But x cannot be a negative number because the log of a negative number is not allowed. So<BR>\n" );
document.write( "log(x) = log(-5) is not allowed in the problem. Therefore, this leaves you with the only<BR>\n" );
document.write( "possible solution of x = +4, and that is the answer to this problem.<BR>\n" );
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document.write( "Check by returning to the original problem and substituting 4 for x to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%284%2B20%29%29+-+log%28%284%2B2%29%29+=+log%284%29 ALIGN=MIDDLE  ALT=\"log%28%284%2B20%29%29+-+log%28%284%2B2%29%29+=+log%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "which simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2824%29+-+log%286%29+=+log%284%29 ALIGN=MIDDLE  ALT=\"log%2824%29+-+log%286%29+=+log%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Use a calculator to find each of these three logs and substitute those values into the<BR>\n" );
document.write( "equation. You will find that the left side of the equation does equal the right side, and <BR>\n" );
document.write( "therefore, the answer of x = +4 checks.<BR>\n" );
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document.write( "Hope this helps you to see your way through the problem.<BR>\n" );
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