document.write( "Question 164034: I'm looking for an explicit solution, not an approximation using iteration.
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\n" ); document.write( "x^x = 2
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Algebra.Com's Answer #120872 by Edwin McCravy(20086)\"\" \"About 
You can put this solution on YOUR website!
I'm looking for an explicit solution, not an approximation using iteration.
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\n" ); document.write( "x^x = 2
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document.write( "Variable exponents are not algebraic.  There would be a\r\n" );
document.write( "Taylor series solution for it.  But there is no algebraic\r\n" );
document.write( "method.  However, just as was done in the case of all the\r\n" );
document.write( "standard transcendental functions such as \r\n" );
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document.write( "f(x) = sin(x), ln(x), exp(x), sinh(x),, etc.\r\n" );
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document.write( "a table of those were made up and printed in a book.  Then \r\n" );
document.write( "mathematicians were able to define things algebraically in \r\n" );
document.write( "terms of those non-algebraic (trancendental) functions. So \r\n" );
document.write( "a standard non-algebraic function could be derived, say, \r\n" );
document.write( "\"floop(x)\", and this could become as widely an accepted \r\n" );
document.write( "standard function as, say, sin(x), which could be defined \r\n" );
document.write( "as the sum of an infinite series. Then the solution to\r\n" );
document.write( "x^x=2 could be defined in terms of floop(x).\r\n" );
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document.write( "But, let's face it, every standard trancendental function,\r\n" );
document.write( "such as sin(x), ln(x), etc. can only be approximated by \r\n" );
document.write( "iterations.  \r\n" );
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document.write( "The solution to your equation is non-algebraic, and can only \r\n" );
document.write( "be calculated by iterative methods.          \r\n" );
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document.write( "Edwin
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