document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196149>Question 1196149</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the exact value of x for which 4cosh(x) - e^-(x) = 3<BR>\n" );
document.write( "Find the exact value of cosh^-¹(3) - sinh^-1(2)<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 #828869 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=828869\">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=4cosh%28x%29+-+e%5E%28-x%29+=+3 ALIGN=MIDDLE  ALT=\"4cosh%28x%29+-+e%5E%28-x%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >............................use identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cosh%28x%29+=e%5E%28-x%29%2F2+%2B+e%5Ex%2F2 ALIGN=MIDDLE  ALT=\"cosh%28x%29+=e%5E%28-x%29%2F2+%2B+e%5Ex%2F2\"   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=4%28e%5E%28-x%29%2F2+%2B+e%5Ex%2F2%29-+e%5E%28-x%29+=+3 ALIGN=MIDDLE  ALT=\"4%28e%5E%28-x%29%2F2+%2B+e%5Ex%2F2%29-+e%5E%28-x%29+=+3\"   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=2e%5E%28-x%29+%2B+2+e%5Ex-e%5E%28-x%29+=3 ALIGN=MIDDLE  ALT=\"2e%5E%28-x%29+%2B+2+e%5Ex-e%5E%28-x%29+=3\"   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%28-x%29+%2B+2e%5Ex+=3 ALIGN=MIDDLE  ALT=\"+e%5E%28-x%29+%2B+2e%5Ex+=3\"   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%28-x%29+%282+e%5E%282+x%29+%2B+1%29+=3 ALIGN=MIDDLE  ALT=\"e%5E%28-x%29+%282+e%5E%282+x%29+%2B+1%29+=3\"   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=2e%5E%282+x%29+%2B+1=3%2Fe%5E%28-x%29+ ALIGN=MIDDLE  ALT=\"2e%5E%282+x%29+%2B+1=3%2Fe%5E%28-x%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=2e%5E%282x%29+%2B+1=3e%5Ex ALIGN=MIDDLE  ALT=\"2e%5E%282x%29+%2B+1=3e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2e%5E%282x%29+-3e%5Ex%2B1=0 ALIGN=MIDDLE  ALT=\"2e%5E%282x%29+-3e%5Ex%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Rewrite the equation  with:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex=u ALIGN=MIDDLE  ALT=\"e%5Ex=u\"   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=2u%5E2-3u=-1 ALIGN=MIDDLE  ALT=\"2u%5E2-3u=-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=2u%5E2-3u%2B1=0 ALIGN=MIDDLE  ALT=\"2u%5E2-3u%2B1=0\"   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=%282u+-+1%29%28u+-+1%29=0 ALIGN=MIDDLE  ALT=\"%282u+-+1%29%28u+-+1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "solutions for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282+u+-+1%29+=0+ ALIGN=MIDDLE  ALT=\"%282+u+-+1%29+=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2u=1 ALIGN=MIDDLE  ALT=\"+2u=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > =><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=1%2F2 ALIGN=MIDDLE  ALT=\"u=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "if  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28u+-+1%29=0 ALIGN=MIDDLE  ALT=\"+%28u+-+1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > =><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=1 ALIGN=MIDDLE  ALT=\"u=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Substitute back <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=e%5Ex ALIGN=MIDDLE  ALT=\"u=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >, solve for<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x ALIGN=MIDDLE  ALT=\"+x\"   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=u=1%2F2 ALIGN=MIDDLE  ALT=\"u=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex=1%2F2+ ALIGN=MIDDLE  ALT=\"e%5Ex=1%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >....take natural log of both sides\r<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%281%2F2%29 ALIGN=MIDDLE  ALT=\"ln%28e%5Ex%29=ln%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%2Aln%28e%29=ln%281%2F2%29 ALIGN=MIDDLE  ALT=\"+x%2Aln%28e%29=ln%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A1=ln%281%29-ln%282%29 ALIGN=MIDDLE  ALT=\"x%2A1=ln%281%29-ln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0-ln%282%29 ALIGN=MIDDLE  ALT=\"x=0-ln%282%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=x+=+-ln%282%29 ALIGN=MIDDLE  ALT=\"x+=+-ln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=1 ALIGN=MIDDLE  ALT=\"u=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex=1 ALIGN=MIDDLE  ALT=\"e%5Ex=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%5Ex%29=ln%281%29 ALIGN=MIDDLE  ALT=\"ln%28e%5Ex%29=ln%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Aln%28e%29=0 ALIGN=MIDDLE  ALT=\"x%2Aln%28e%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A1=0 ALIGN=MIDDLE  ALT=\"x%2A1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "verify solutions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4cosh%28-ln%282%29%29+-+e%5E%28-%28-ln%282%29%29%29+=+3 ALIGN=MIDDLE  ALT=\"4cosh%28-ln%282%29%29+-+e%5E%28-%28-ln%282%29%29%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%285%2F4%29+-+2%29+=+3 ALIGN=MIDDLE  ALT=\"4%285%2F4%29+-+2%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+-+2%29+=+3 ALIGN=MIDDLE  ALT=\"5+-+2%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%29+=+3 ALIGN=MIDDLE  ALT=\"3%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > => true\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4cosh%280%29+-+e%5E%28-0%29+=+3 ALIGN=MIDDLE  ALT=\"4cosh%280%29+-+e%5E%28-0%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%281%29+-+1+=+3 ALIGN=MIDDLE  ALT=\"4%281%29+-+1+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3=+3 ALIGN=MIDDLE  ALT=\"3=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cosh%5E-1%283%29+-+sinh%5E-1%282%29 ALIGN=MIDDLE  ALT=\"cosh%5E-1%283%29+-+sinh%5E-1%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Use the following identities: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sinh%5E-1%28x%29=ln%28sqrt%281%2Bx%5E2%29%2Bx%29 ALIGN=MIDDLE  ALT=\"sinh%5E-1%28x%29=ln%28sqrt%281%2Bx%5E2%29%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cosh%5E-1%28x%29=ln%28sqrt%28-1%2Bx%5E2%29%2Bx%29 ALIGN=MIDDLE  ALT=\"cosh%5E-1%28x%29=ln%28sqrt%28-1%2Bx%5E2%29%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28sqrt%28-1%2B3%5E2%29%2B3%29-+ln%28sqrt%281%2B2%5E2%29%2B2%29 ALIGN=MIDDLE  ALT=\"ln%28sqrt%28-1%2B3%5E2%29%2B3%29-+ln%28sqrt%281%2B2%5E2%29%2B2%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=ln%282sqrt%282%29+%2B+3%29-+ln+%28sqrt%285%29+%2B+2%29 ALIGN=MIDDLE  ALT=\"ln%282sqrt%282%29+%2B+3%29-+ln+%28sqrt%285%29+%2B+2%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=ln%28%282sqrt%282%29+%2B+3%29%2F%28sqrt%285%29+%2B+2%29%29+ ALIGN=MIDDLE  ALT=\"ln%28%282sqrt%282%29+%2B+3%29%2F%28sqrt%285%29+%2B+2%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> exact solution\r<BR>\n" );
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document.write( "  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );