document.write( "Question 694069: log(x-3)+log(x) = 2 + log(x+2) *The base for the logarithms is 2. I just need help clarifying my answer, I found that this problem had no solution but I'm not completely sure if that is correct \n" ); document.write( "
Algebra.Com's Answer #427729 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "No, it has a solution.\r\n" );
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document.write( "The four rules of logarithms are:\r\n" );
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document.write( "1.  \"log%28b%2C%28A%29%29%2Blog%28b%2C%28C%29%29=log%28b%2C%28AC%29%29\"\r\n" );
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document.write( "2.  \"log%28b%2C%28A%29%29-log%28b%2C%28C%29%29=log%28b%2C%28A%2FC%29%29\"\r\n" );
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document.write( "3.  \"log%28b%2C%28A%2AC%29%29=C%2Alog%28b%2C%28A%29%29\"\r\n" );
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document.write( "4.  log equation \"log%28b%2C%28A%29%29=C\" is equivalent to\r\n" );
document.write( "exponential equation \"A=b%5EC\"\r\n" );
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document.write( "5.  \"log%28b%2C%28n%29%29=1\"\r\n" );
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document.write( "       log2(x-3) + log2(x) = 2 + log2(x+2)\r\n" );
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document.write( "Use rule 1 on the left side:\r\n" );
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document.write( "              log2[(x-3)x] = 2 + log2(x+2)\r\n" );
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document.write( "Distribute\r\n" );
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document.write( "               log2(x²-3x) = 2 + log2(x+2)\r\n" );
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document.write( "Get bothlogs on the left side of the equation:\r\n" );
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document.write( "   log2(x²-3x) - log2(x+2) = 2\r\n" );
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document.write( "Use rule 2 on the left\r\n" );
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document.write( "              log2\"%28%28x%5E2-3x%29%2F%28x%2B2%29%29\" = 2\r\n" );
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document.write( "Use rule 4\r\n" );
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document.write( "               \"%28x%5E2-3x%29%2F%28x%2B2%29\" = 22\r\n" );
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document.write( "Write 2² as 4\r\n" );
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document.write( "              \"%28x%5E2-3x%29%2F%28x%2B2%29\" = 4\r\n" );
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document.write( "Multiply both sides by x+2\r\n" );
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document.write( "              x² - 3x = 4(x + 2)\r\n" );
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document.write( "              x² - 3x = 4x + 8\r\n" );
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document.write( "          x² - 7x - 8 = 0\r\n" );
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document.write( "       (x - 8)(x + 1) = 0\r\n" );
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document.write( "       x - 8 = 0;  x + 1 = 0\r\n" );
document.write( "           x = 8;      x = -1\r\n" );
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document.write( "The only solution is x = 8.  Logarithms of negative\r\n" );
document.write( "numbers are not real numbers.  So we discard the\r\n" );
document.write( "negative answer.\r\n" );
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document.write( "Checking:\r\n" );
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document.write( "       log2(x-3) + log2(x) = 2 + log2(x+2)\r\n" );
document.write( "       log2(8-3) + log2(8) = 2 + log2(8+2)\r\n" );
document.write( "        log2(5) + log2(2³) = 2 + log2(10)\r\n" );
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document.write( "Use rule 3 on 2nd term, Write 10 as 2·5\r\n" );
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document.write( "       log2(5) + 3·log2(2) = 2 + log2(2·5)\r\n" );
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document.write( "Use rule 5 to rewrite log2(2) as 1\r\n" );
document.write( "Use rule 1 to rewrite last term on right\r\n" );
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document.write( "       log2(5) + 3·1 = 2 + log2(2) + log2(5)\r\n" );
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document.write( "Use rule 5 to rewrite log2(2) as 1\r\n" );
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document.write( "         log2(5) + 3 = 2 + 1 + log2(5)\r\n" );
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document.write( "Add 2 + 1\r\n" );
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document.write( "         log2(5) + 3 = 3 + log2(5)\r\n" );
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document.write( "So it checks.\r\n" );
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document.write( "Edwin
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