document.write( "Question 1153084: The product of all the real roots of x^log(base 10)x = 10 is \n" ); document.write( "
Algebra.Com's Answer #775259 by ikleyn(53763)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "You are given this equation\r\n" );
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document.write( "    \"x%5Elog%2810%2C%28x%29%29\" = 10.\r\n" );
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document.write( "Take log(base 10) from both sides.  You will get\r\n" );
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document.write( "    \"log%2810%2C%28x%29%29%2Alog%2810%2C%28x%29%29\" = 1,\r\n" );
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document.write( "or\r\n" );
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document.write( "    \"%28log%2810%2C%28x%29%29%29%5E2\" = 1.\r\n" );
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document.write( "Take the square root from both sides\r\n" );
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document.write( "    \"log%2810%2C%28x%29%29\" = +/- 1.\r\n" );
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document.write( "Case 1.   \"log%2810%2C%28x%29%29\" = 1.\r\n" );
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document.write( "          It implies  x = 10.\r\n" );
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document.write( "Case 2.  \"log%2810%2C%28x%29%29\" = -1.\r\n" );
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document.write( "         It implies  x = \"1%2F10\".\r\n" );
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document.write( "ANSWER.  The given equation has 2 (two, TWO) solutions  x = 10  and  x = \"1%2F10\".\r\n" );
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document.write( "      +------------------------------------------------------------------+\r\n" );
document.write( "      | The product of the roots (as the question asks) is  10*\"1%2F10\" = 1.   |\r\n" );
document.write( "      +------------------------------------------------------------------+\r\n" );
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document.write( "To check, I made two plots with different x-scales of the function  y = \"x%5Elog%2810%2C%28x%29%29\".  See below.\r\n" );
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document.write( "Plot y = \"x%5Elog%2810%2C%28x%29%29\" (red) and y = 10 (green).\r\n" );
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document.write( "Plot y = \"x%5Elog%2810%2C%28x%29%29\" (red) and y = 10 (green).\r\n" );
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\n" ); document.write( "\n" ); document.write( "In the post by  @MathLover1,  the solution   x = \"1%2F10\"   is lost.\r
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