document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=484241>Question 484241</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm having trouble with this problem. Thanks for your help.<BR>\n" );
document.write( "Solve. Express x as a logarithm if necessary.<BR>\n" );
document.write( "e^(x)+e^(-x)=4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #331339 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=331339\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28x%29%2Be%5E%28-x%29=4 ALIGN=MIDDLE  ALT=\"e%5E%28x%29%2Be%5E%28-x%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28x%29%2B1%2F%28e%5E%28x%29%29=4 ALIGN=MIDDLE  ALT=\"e%5E%28x%29%2B1%2F%28e%5E%28x%29%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-x%29 ALIGN=MIDDLE  ALT=\"e%5E%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28e%5E%28x%29%29 ALIGN=MIDDLE  ALT=\"1%2F%28e%5E%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x%29%2B1=4e%5Ex ALIGN=MIDDLE  ALT=\"e%5E%282x%29%2B1=4e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply EVERY term by the LCD <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 clear out the fractions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x%29%2B1-4e%5Ex=0 ALIGN=MIDDLE  ALT=\"e%5E%282x%29%2B1-4e%5Ex=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Get everything to one side.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x%29-4e%5Ex%2B1=0 ALIGN=MIDDLE  ALT=\"e%5E%282x%29-4e%5Ex%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28e%5Ex%29%5E2-4e%5Ex%2B1=0 ALIGN=MIDDLE  ALT=\"%28e%5Ex%29%5E2-4e%5Ex%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite the first term\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2-4z%2B1=0 ALIGN=MIDDLE  ALT=\"z%5E2-4z%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace EVERY copy of <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;\" > with 'z' (so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=e%5Ex ALIGN=MIDDLE  ALT=\"z=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Notice that the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E2-4z%2B1 ALIGN=MIDDLE  ALT=\"z%5E2-4z%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Az%5E2%2BBz%2BC ALIGN=MIDDLE  ALT=\"Az%5E2%2BBz%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1 ALIGN=MIDDLE  ALT=\"A=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-4 ALIGN=MIDDLE  ALT=\"B=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=1 ALIGN=MIDDLE  ALT=\"C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for \"z\":\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 ALIGN=MIDDLE  ALT=\"z+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-%28-4%29+%2B-+sqrt%28+%28-4%29%5E2-4%281%29%281%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-%28-4%29+%2B-+sqrt%28+%28-4%29%5E2-4%281%29%281%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1 ALIGN=MIDDLE  ALT=\"A=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-4 ALIGN=MIDDLE  ALT=\"B=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=1 ALIGN=MIDDLE  ALT=\"C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%284+%2B-+sqrt%28+%28-4%29%5E2-4%281%29%281%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%284+%2B-+sqrt%28+%28-4%29%5E2-4%281%29%281%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%284+%2B-+sqrt%28+16-4%281%29%281%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%284+%2B-+sqrt%28+16-4%281%29%281%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%284+%2B-+sqrt%28+16-4+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%284+%2B-+sqrt%28+16-4+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%281%29%281%29 ALIGN=MIDDLE  ALT=\"4%281%29%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%284+%2B-+sqrt%28+12+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"z+=+%284+%2B-+sqrt%28+12+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%284+%2B-+sqrt%28+12+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%284+%2B-+sqrt%28+12+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%284+%2B-+2%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%284+%2B-+2%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%284%29%2F%282%29+%2B-+%282%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"z+=+%284%29%2F%282%29+%2B-+%282%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction.  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+2+%2B-+sqrt%283%29 ALIGN=MIDDLE  ALT=\"z+=+2+%2B-+sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+2%2Bsqrt%283%29 ALIGN=MIDDLE  ALT=\"z+=+2%2Bsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+2-sqrt%283%29 ALIGN=MIDDLE  ALT=\"z+=+2-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression.  \r<BR>\n" );
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document.write( "So the solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+2%2Bsqrt%283%29 ALIGN=MIDDLE  ALT=\"z+=+2%2Bsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+2-sqrt%283%29 ALIGN=MIDDLE  ALT=\"z+=+2-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "But remember, we said that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=e%5Ex ALIGN=MIDDLE  ALT=\"z=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex+=+2%2Bsqrt%283%29 ALIGN=MIDDLE  ALT=\"e%5Ex+=+2%2Bsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex+=+2-sqrt%283%29 ALIGN=MIDDLE  ALT=\"e%5Ex+=+2-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now convert to logarithmic form to get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+ln%282%2Bsqrt%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+ln%282%2Bsqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+ln%282-sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+ln%282-sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the solutions are \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+ln%282%2Bsqrt%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+ln%282%2Bsqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+ln%282-sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"x+=+ln%282-sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );