.
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document.write( "Find x, 
= 
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document.write( "Your starting equation is \r\n" );
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document.write( " = . (1)\r\n" );
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document.write( "Take natural logarithm of both sides\r\n" );
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document.write( " 5*ln(x) = x*ln(9).\r\n" );
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document.write( "Divide both sides by ln(x)*ln(9). You will get an EQUIVALENT equation\r\n" );
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document.write( " 
= 
. (2)\r\n" );
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document.write( "It is well known fact that the function has the minimum at x = e,\r\n" );
document.write( "where e is the base of natural logarithms, e = 2.71828...\r\n" );
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document.write( "The minimum of this function at x= e is 
= 
= e = 2.71828...\r\n" );
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document.write( "From the other side, the value of is 2.275598067...\r\n" );
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document.write( "THEREFORE, equation (2) has NO solution at x > 0.\r\n" );
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document.write( "ANSWER. In domain x > 0, the given equation (1) has no solution.\r\n" );
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document.write( "Solved.\r
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document.write( "In order to convince yourself visually with the fact that equation (2)
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document.write( "has no solutions, go to website www.desmos.com/calculator.\r
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document.write( "It allows to plot functions for free automatically.\r
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document.write( "Print the expression y = , get the plot and compare it with the plot
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document.write( "of horizontal line y = .\r
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document.write( "=====================\r
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document.write( "Post-solution note\r
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document.write( " This trick with using monotonicity of the function is a powerful tool for solving\r\n" );
document.write( " many/some exponential-polynomial equations, similar to the given in this post\r\n" );
document.write( " or for proving that such equations have no solutions.\r\n" );
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document.write( " When nothing else does not work, try it . . . \r\n" );
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
