document.write( "Question 1210406: 8^x=16^x+4 \n" ); document.write( "

Algebra.Com's Answer #852252 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "8^x = 16^x + 4
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document.write( "I read this equation exactly and literally as it is written\r\n" );
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document.write( "     =  + .     (1)\r\n" );
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document.write( "My statement is that this equation has no solutions in real numbers.\r\n" );
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document.write( "Indeed, if  x >= 0,  then, obviously,    is less than  ;\r\n" );
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document.write( "Hence, a fortiori,   <  + ,  so equation (1) has no solutions in the domain  x >= 0.\r\n" );
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document.write( "Next, if  x < 0,  then left side of equation (1) is less than 1,\r\n" );
document.write( "while right side of this equation is greater than 4, so equation (1) has no solutions in the domain   x < 0.\r\n" );
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document.write( "Thus the statement is proved and the equation has no solutions in real domain.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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document.write( "If, in opposite, your original equation is\r\n" );
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document.write( "     = ,    (2)\r\n" );
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document.write( "then the reasoning is different and the answer is different, too.\r\n" );
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document.write( "Then this equation (2) can be rewritten equivalently in base '2'\r\n" );
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document.write( "     = ,\r\n" );
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document.write( "which implies an equation for indexes\r\n" );
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document.write( "    3x = 4*(x+4),\r\n" );
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document.write( "    3x = 4x + 16,\r\n" );
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document.write( "    3x - 4x = 16,\r\n" );
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document.write( "       -x   = 16,\r\n" );
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document.write( "        x   = -16,\r\n" );
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document.write( "so the solution for equation (2) is x = -16.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved two times for two different interpretations.\r
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\n" ); document.write( "\n" ); document.write( "By the way, both interpretations lead to meaningful and instructive solutions.\r
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