document.write( "Question 1209562: (7^log₈x)(x^log₉x) = 3969,
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Algebra.Com's Answer #849531 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "With a TI-84 and other technology it is easy to get an approximate answer.\r\n" );
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document.write( "An approximate answer I can read right off my trusty TI-84 calculator as\r\n" );
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document.write( "x = 15.545914\r\n" );
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document.write( "However, the teacher who assigned this problem is probably not teaching a course\r\n" );
document.write( "on how to use technology to solve math problems.  So I think students who are\r\n" );
document.write( "assigned problems like this are supposed to get EXACT answers, in terms of\r\n" );
document.write( "standard functions, such as logarithms and exponential functions. So I'll strive\r\n" );
document.write( "for an EXACT answer. It's not going to be easy.   \r\n" );
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document.write( "(7^log₈x)(x^log₉x) = 3969, \r\n" );
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document.write( "\"%287%5Elog%283%2C%28x%29%29%29%28x%5Elog%289%2C%28x%29%29%29=3969\"\r\n" );
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document.write( "The left side is a strictly increasing function.\r\n" );
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document.write( "When x=9 the left side is\r\n" );
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document.write( "\"%287%5Elog%283%2C%289%29%29%29%2A%289%5Elog%289%2C%289%29%29%29\"\r\n" );
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document.write( "\"7%5E2%289%5E1%29=441\"\r\n" );
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document.write( "Thus x must be greater than 9, since the left side must increase all the way to\r\n" );
document.write( "3969.\r\n" );
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document.write( "\"%287%5Elog%283%2C%28x%29%29%29%28x%5Elog%289%2C%28x%29%29%29=3969\"\r\n" );
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document.write( "\"%287%5Elog%283%2C%28x%29%29%29%28x%5Elog%289%2C%28x%29%29%29=7%5E2%2A3%5E4\"\r\n" );
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document.write( "Divide both sides by 72.\r\n" );
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document.write( "\"%287%5E%28log%283%2C%28x%29%29-2%29%29%28x%5Elog%289%2C%28x%29%29%29=3%5E4\" \r\n" );
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document.write( "Take natural logs of both sides:\r\n" );
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document.write( "\"ln%287%5E%28log%283%2C%28x%29%29-2%29%29%2Bln%28x%5Elog%289%2C%28x%29%29%29\"\"%22%22=%22%22\"\"4%2Aln%283%29\"\r\n" );
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document.write( "\"%28log%283%2C%28x%29%29-2%29ln%287%29%2B%28log%289%2C%28x%29%29%29ln%28x%29\"\"%22%22=%22%22\"\"4%2Aln%283%29\"\r\n" );
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document.write( "Change to natural logs:\r\n" );
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document.write( "\"%28ln%28x%29%2Fln%283%29-2%29ln%287%29%2B%28ln%28x%29%2Fln%289%29%29ln%28x%29\"\"%22%22=%22%22\"\"4%2Aln%283%29\"\r\n" );
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document.write( "Since \"ln%289%29=ln%283%5E2%29=2%2Aln%283%29\",\r\n" );
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document.write( "\"%28ln%28x%29%2Fln%283%29-2%29ln%287%29%2B%28ln%28x%29%2F%282%2Aln%283%29%29%29ln%28x%29\"\"%22%22=%22%22\"\"4%2Aln%283%29\"\r\n" );
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document.write( "let ln(x) = y,  ln(3) = a, and ln(7) = b\r\n" );
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document.write( "\"%28y%2Fa-2%29%2Ab%2B%28y%2F%282a%29%29y\"\"%22%22=%22%22\"\"4a\"\r\n" );
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document.write( "\"%28by%29%2Fa-2b%2By%5E2%2F%282a%5E%22%22%29\"\"%22%22=%22%22\"\"4a\"\r\n" );
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document.write( "\"2by-4ab%2By%5E2\"\"%22%22=%22%22\"\"8a%5E2\"\r\n" );
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document.write( "\"y%5E2%2B2by-4ab-8a%5E2\"\"%22%22=%22%22\"\"0\"\r\n" );
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document.write( "\"y%5E2%2B2by-4a%28b%2B2a%29=0\"\r\n" );
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document.write( "\"y+=+%28-2b+%2B-+sqrt%28+%282b%29%5E2-4%2A1%2A%28-4a%28b%2B2a%29%29%29+%29%2F%282%2A1%29+\"\r\n" );
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document.write( "\"y+=+%28-2b+%2B-+sqrt%28+4b%5E2%2B16a%28b%2B2a%29%29+%29%2F2+\"\r\n" );
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document.write( "\"y+=+%28-2b+%2B-+sqrt%28+4%28b%5E2%2B4a%28b%2B2a%29%29%29+%29%2F2+\"\r\n" );
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document.write( "\"y+=+%28-2b+%2B-+2sqrt%28+b%5E2%2B4ab%2B8a%5E2%29%29+%2F2+\"\r\n" );
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document.write( "\"y+=+-b+%2B-+sqrt%28b%5E2%2B4ab%2B8a%5E2%29++\"\r\n" );
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document.write( "\"ln%28x%29+=+-ln%287%29+%2B-+sqrt%28ln%287%29%5E2%2B4ln%283%29ln%287%29%2B8ln%283%29%5E2%29++\"\r\n" );
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document.write( "Since x > 9 then ln(x) > 2, so we must use the + sign.\r\n" );
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document.write( "\"x\"\"%22%22=%22%22\"\r\n" );
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document.write( "\"x\"\"%22%22=%22%22\"\r\n" );
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document.write( "Since \"e%5E%28-ln%287%29%29=1%5E%22%22%2Fe%5E%28ln%287%29%29=1%2F7\"\r\n" );
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document.write( "\"x\"\"%22%22=%22%22\"\r\n" );
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document.write( "That's the EXACT answer.\r\n" );
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document.write( "When you evaluate that with a calculator, you'll get x = 15.54591432.\r\n" );
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document.write( "Edwin
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