document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1106986>Question 1106986</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. x^x=46656<BR>\n" );
document.write( "2. x!=3628800<BR>\n" );
document.write( "If I plug in some values, I can solve, but the problem is that I do not figure out a way to list out the possess. So what's the possess? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #721998 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=721998\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Are you by chance studying a chapter about estimation and/or iterative solutions?   Both these problems lend themselves to iterative (\"guessing\") methods.<BR>\n" );
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document.write( "1.   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5Ex+=+46656+ ALIGN=MIDDLE  ALT=\"+x%5Ex+=+46656+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the left hand side changes rapidly with changes in x,  I chose to re-write it by taking ln() of both sides:<BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%2Aln%28x%29+=+ln%2846656%29+ ALIGN=MIDDLE  ALT=\"+x%2Aln%28x%29+=+ln%2846656%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > approx= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+10.7505568+ ALIGN=MIDDLE  ALT=\"+10.7505568+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now the left side doesn't change very rapidly so as we get close the guesses won't change much:\r<BR>\n" );
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document.write( "   4 ln(4)  approx= 5.545   (way too low)<BR>\n" );
document.write( "   5 ln(5)  approx= 8.047 <BR>\n" );
document.write( "   6 ln(6)  approx= 10.7505568\r<BR>\n" );
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document.write( "Note that this is exactly the same estimate we are looking for, checking <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%5E6 ALIGN=MIDDLE  ALT=\"6%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > shows it is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+46656+ ALIGN=MIDDLE  ALT=\"+46656+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  so the answer here is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28+x=6+%29 ALIGN=MIDDLE  ALT=\"+highlight%28+x=6+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.   Note that the guessing method would have worked even if x was a real number, not just an integer. \r<BR>\n" );
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document.write( "—<BR>\n" );
document.write( "2.  There is no easy inverse function for the factorial (gamma function).<BR>\n" );
document.write( "However, given x! = 3628800  you can start by dividing by 2, then 3, then 4, etc.  When your division produces a value of 1 then that last division is the value of x.\r<BR>\n" );
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document.write( "Example:  n! = 24<BR>\n" );
document.write( "             n/2 = 12<BR>\n" );
document.write( "             12/3 = 4<BR>\n" );
document.write( "              4/4 = 1    <<<  4! = 24<BR>\n" );
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document.write( "Another approach is to iterate (guess, and adjust your next guess up or down accordingly):<BR>\n" );
document.write( "    8! = 40320             (too low, try a larger value of x)<BR>\n" );
document.write( "   12! = 479001600    (too high, next try a smaller value of x)<BR>\n" );
document.write( "So we already know  8 < x < 12,   there are only 3 values left to try.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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