document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=120725>Question 120725</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve: \r<BR>\n" );
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document.write( "a) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+log%282%5Ex%29+=+2+%5E%28logx%29+ ALIGN=MIDDLE  ALT=\"+log%282%5Ex%29+=+2+%5E%28logx%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+e%5Ex+%2B+e+%5E%28-x%29+=+6+ ALIGN=MIDDLE  ALT=\"+e%5Ex+%2B+e+%5E%28-x%29+=+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #88532 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=88532\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> These are a couple of complex problems. It might get a little tricky to follow the math. I'm <BR>\n" );
document.write( "going to presume that I don't need to explain the details of each rule of logs that are<BR>\n" );
document.write( "used. And I'm going to presume the logs are to the base 10 ... <BR>\n" );
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document.write( "(a) Given<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%5Ex%29+=+2%5E%28log%28x%29%29 ALIGN=MIDDLE  ALT=\"log%282%5Ex%29+=+2%5E%28log%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The exponent on the left side can be brought out as a multiplier to give:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Alog%282%29+=+2%5E%28log%28x%29%29 ALIGN=MIDDLE  ALT=\"x%2Alog%282%29+=+2%5E%28log%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Take the log of both sides:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%2Alog%282%29%29+=+log%282%5E%28log%28x%29%29%29 ALIGN=MIDDLE  ALT=\"log%28x%2Alog%282%29%29+=+log%282%5E%28log%28x%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Apply the exponential rule to the right side:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%2Alog%282%29%29+=+log%28x%29%2Alog%282%29 ALIGN=MIDDLE  ALT=\"log%28x%2Alog%282%29%29+=+log%28x%29%2Alog%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Split the left side into two logs using the multiplication rule:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%29%2Blog%28log%282%29%29=log%28x%29%2Alog%282%29 ALIGN=MIDDLE  ALT=\"log%28x%29%2Blog%28log%282%29%29=log%28x%29%2Alog%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28log%282%29%29 ALIGN=MIDDLE  ALT=\"log%28log%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left side by subtracting it from both sides to result in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%29=log%28x%29%2Alog%282%29-log%28log%282%29%29 ALIGN=MIDDLE  ALT=\"log%28x%29=log%28x%29%2Alog%282%29-log%28log%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%29%2Alog%282%29 ALIGN=MIDDLE  ALT=\"log%28x%29%2Alog%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the right side by subtracting it from both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%29-log%28x%29%2Alog%282%29-log%28log%282%29%29 ALIGN=MIDDLE  ALT=\"log%28x%29-log%28x%29%2Alog%282%29-log%28log%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "On the left side factor out the common term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%29 ALIGN=MIDDLE  ALT=\"log%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%29%281-log%282%29%29=-log%28log%282%29%29 ALIGN=MIDDLE  ALT=\"log%28x%29%281-log%282%29%29=-log%28log%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%29 ALIGN=MIDDLE  ALT=\"log%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by dividing both sides of this equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-log%282%29 ALIGN=MIDDLE  ALT=\"1-log%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%29+=+-log%28log%282%29%29%2F%281-log%282%29%29 ALIGN=MIDDLE  ALT=\"log%28x%29+=+-log%28log%282%29%29%2F%281-log%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now it's calculator time ... on the right side calculate the numerator by entering 2 and <BR>\n" );
document.write( "taking the base 10 logarithm twice in succession. You should get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.521390227 ALIGN=MIDDLE  ALT=\"-0.521390227\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "as the answer, but don't forget the minus sign preceding the numerator ... so the numerator<BR>\n" );
document.write( "is positive and is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.521390227 ALIGN=MIDDLE  ALT=\"0.521390227\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The denominator is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-log%282%29 ALIGN=MIDDLE  ALT=\"1-log%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a calculator will<BR>\n" );
document.write( "tell you that that computes to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.698970004 ALIGN=MIDDLE  ALT=\"0.698970004\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This reduces the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%29+=+0.521390227%2F0.698970004=+0.745940776 ALIGN=MIDDLE  ALT=\"log%28x%29+=+0.521390227%2F0.698970004=+0.745940776\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So we have that:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28x%29=0.745940776 ALIGN=MIDDLE  ALT=\"log%28x%29=0.745940776\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "To solve for x convert from this logarithmic form to the exponential form as follows:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+10%5E0.745940776=+5.57109772 ALIGN=MIDDLE  ALT=\"x+=+10%5E0.745940776=+5.57109772\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and that's the answer ... or as close to the correct answer as my calculator allows.<BR>\n" );
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document.write( "Note that x = 0 comes pretty close to being a correct answer also, but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%280%29 ALIGN=MIDDLE  ALT=\"log%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > will<BR>\n" );
document.write( "give you an error when you try to compute it. If you let x approach zero from the positive<BR>\n" );
document.write( "side your given equation will become very close to equal on both sides. You can see this by<BR>\n" );
document.write( "letting x be something such as 1 times 10^(-99) ... and plugging that very small value into <BR>\n" );
document.write( "the given equation.<BR>\n" );
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document.write( "Next problem (b) Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+e%5Ex+%2B+e+%5E%28-x%29+=+6+ ALIGN=MIDDLE  ALT=\"+e%5Ex+%2B+e+%5E%28-x%29+=+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply all terms on both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+e%5E2x+%2B+e+%5E%280%29+=+6%2Ae%5Ex+ ALIGN=MIDDLE  ALT=\"+e%5E2x+%2B+e+%5E%280%29+=+6%2Ae%5Ex+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E0+=+1 ALIGN=MIDDLE  ALT=\"e%5E0+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E2x%2B1=6%2Ae%5Ex ALIGN=MIDDLE  ALT=\"e%5E2x%2B1=6%2Ae%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2Ae%5Ex ALIGN=MIDDLE  ALT=\"6%2Ae%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides. The resulting equation is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E2x+-6%2Ae%5Ex%2B1=0 ALIGN=MIDDLE  ALT=\"e%5E2x+-6%2Ae%5Ex%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Just to simplify the problem a little, let's let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+e%5Ex ALIGN=MIDDLE  ALT=\"y+=+e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If we substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2B6y%2B1=0 ALIGN=MIDDLE  ALT=\"y%5E2%2B6y%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is an \"ordinary\" quadratic equation of standard form. Apply the quadratic formula to solve<BR>\n" );
document.write( "it and you should get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-6+%2B-+sqrt%28+%28-6%29%5E2-4%2A1%2A1+%29%29%2F%282%29=%286%2B-sqrt%2836-4%29%29%2F2=3%2B-2sqrt%282%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Using a calculator you should find that the two answers are:<BR>\n" );
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document.write( "y = 5.828427125 and y = 0.171572875<BR>\n" );
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document.write( "But recall that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=e%5Ex ALIGN=MIDDLE  ALT=\"y=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the two answers become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex+=+5.828427125 ALIGN=MIDDLE  ALT=\"e%5Ex+=+5.828427125\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex+=+0.171572875 ALIGN=MIDDLE  ALT=\"e%5Ex+=+0.171572875\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Take the natural log of both sides. For the first answer you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%5Ex%29+=+ln%285.828427125%29 ALIGN=MIDDLE  ALT=\"ln%28e%5Ex%29+=+ln%285.828427125%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The exponent on the left side becomes a multiplier and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Aln%28e%29+=+ln%285.828427125%29 ALIGN=MIDDLE  ALT=\"x%2Aln%28e%29+=+ln%285.828427125%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%29+=+1 ALIGN=MIDDLE  ALT=\"ln%28e%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the equation is reduced to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+ln%285.828427125%29 ALIGN=MIDDLE  ALT=\"x+=+ln%285.828427125%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "and a calculator will tell you that the right side is 1.762747174 so the answer is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1.762747174 ALIGN=MIDDLE  ALT=\"x+=+1.762747174\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If you follow the exact same steps as above, but substitute 0.171572875 in place of 5.828427125, <BR>\n" );
document.write( "you will get that the second answer for x (based on y equaling 0.171572875 this time) is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+ln%280.171572875%29+=+-1.762747174 ALIGN=MIDDLE  ALT=\"x+=+ln%280.171572875%29+=+-1.762747174\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Hope you can make sense of all this mess above and that you can learn something by diligently<BR>\n" );
document.write( "working your way through the labyrinth ... Good luck </SPAN>\n" );
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