document.write( "Question 644781: solve the solution set for x:
\n" ); document.write( "3^(2x+1) - 5*2^(2x+1) = 6^x \n" ); document.write( "
Algebra.Com's Answer #405285 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "This requires these two principles of exponents\r\n" );
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document.write( "(1)    \"a%5E%28b%2Bc%29\" = \"a%5Eb%2Aa%5Ec\"  <--The reversal of adding exponents\r\n" );
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document.write( "(2)    \"%28a%5Eb%29%2F%28c%5Eb%29\" = \"%28a%2Fc%29%5Eb\" <--The reversal of the power of a fraction \r\n" );
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document.write( "\"3%5E%282x%2B1%29\" - \"5%2A2%5E%282x%2B1%29\" = \"6%5Ex\"\r\n" );
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document.write( "Divide through by \"2%5E%282x%2B1%29\"\r\n" );
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document.write( "\"3%5E%282x%2B1%29%2F2%5E%282x%2B1%29\" - 5 = \"6%5Ex%2F%282%5E%282x%2B1%29%29\"\r\n" );
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document.write( "Use principle (2) on the left side\r\n" );
document.write( "Write 6 as 2*3 on the right\r\n" );
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document.write( "\"%283%2F2%29%5E%282x%2B1%29\" - 5 = \"%282%2A3%29%5Ex%2F%282%5E%282x%2B1%29%29\"\r\n" );
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document.write( "Use principle (1) on the right side\r\n" );
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document.write( "\"%283%2F2%29%5E%282x%2B1%29\" - 5 = \"%282%5Ex%2A3%5Ex%29%2F%282%5E%282x%2B1%29%29\"\r\n" );
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document.write( "Use principle (1) on the left\r\n" );
document.write( "Subtract exponents of 2 on the right\r\n" );
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document.write( "\"%283%2F2%29%5E%282x%29%2A%283%2F2%29%5E1%29\" - 5 = \"3%5Ex%2F%282%5E%282x%2B1-x%29%29\"\r\n" );
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document.write( "Simplify\r\n" );
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document.write( "\"%283%2F2%29%5E%282x%29%2A%283%2F2%29%29\" - 5 = \"3%5Ex%2F%282%5E%28x%2B1%29%29\"\r\n" );
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document.write( "Swap factors on the left, use principle (1) on the right denominator\r\n" );
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document.write( "\"%283%2F2%29%283%2F2%29%5E%282x%29%29\" - 5 = \"3%5Ex%2F%282%5Ex%2A2%5E1%29%29\"\r\n" );
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document.write( "Remove the 1 exponent\r\n" );
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document.write( "\"%283%2F2%29%283%2F2%29%5E%282x%29%29\" - 5 = \"3%5Ex%2F%282%5Ex%2A2%29%29\"\r\n" );
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document.write( "Multiply through by 2\r\n" );
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document.write( "\"2%2A%283%2F2%29%283%2F2%29%5E%282x%29%29\" - 2·5 = \"2%2Aexpr%283%5Ex%2F%282%5Ex%2A2%29%29%29\"\r\n" );
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document.write( "\"cross%282%29%2A%283%2Fcross%282%29%29%283%2F2%29%5E%282x%29%29\" - 10 = \"cross%282%29%2Aexpr%283%5Ex%2F%282%5Ex%2Across%282%29%29%29%29\"\r\n" );
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document.write( "\"3%283%2F2%29%5E%282x%29%29\" - 10 = \"3%5Ex%2F2%5Ex\"\r\n" );
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document.write( "\"3%283%2F2%29%5E%282x%29%29\" - 10 = \"%283%2F2%29%5Ex\"\r\n" );
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document.write( "\"3%283%2F2%29%5E%282x%29%29\" - \"%283%2F2%29%5Ex\" - 10 = 0\r\n" );
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document.write( "let u = \"%283%2F2%29%5Ex\"\r\n" );
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document.write( "Then uČ = \"%28%283%2F2%29%5Ex%29%5E2\" = \"%283%2F2%29%5E%282x%29%29\"\r\n" );
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document.write( "Then we have\r\n" );
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document.write( "3uČ - u - 10 = 0\r\n" );
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document.write( "Factoring,\r\n" );
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document.write( "(u - 2)(3u + 5) = 0\r\n" );
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document.write( "Use the zero-factor principle:\r\n" );
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document.write( "u - 2 = 0;  3u + 5 = 0\r\n" );
document.write( "    u = 2       3u = -5\r\n" );
document.write( "                 u = \"-5%2F3\"\r\n" );
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document.write( "Use u = 2\r\n" );
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document.write( "Since u = \"%283%2F2%29%5Ex\"\r\n" );
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document.write( "\"%283%2F2%29%5Ex\" = 2\r\n" );
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document.write( "Take logs of both sides:\r\n" );
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document.write( "\"log%28%28%283%2F2%29%5Ex%29%29\" = \"log%28%282%29%29\"\r\n" );
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document.write( "Use the principle \"log%28%28b%5Ea%29%29=a%2Alog%28%28b%29%29\"\r\n" );
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document.write( "\"x%2Alog%28%283%2F2%29%29\" = \"log%28%282%29%29\"\r\n" );
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document.write( "Divide both sides by \"log%28%283%2F2%29%29\"\r\n" );
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document.write( "\"x\" = \"log%28%282%29%29%2Flog%28%283%2F2%29%29\" = 1.709511291\r\n" );
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document.write( "That's the only solution because if we try \r\n" );
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document.write( "u = \"-5%2F3\"\r\n" );
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document.write( "\"%283%2F2%29%5Ex\" = \"-5%2F3\"\r\n" );
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document.write( "That cannot produce an additional solution because because \"3%2F2\" is\r\n" );
document.write( "positive and a positive number raised to any power, positive or negative,\r\n" );
document.write( "gives a positive answer. \r\n" );
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document.write( "Edwin
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