document.write( "Question 221046: how do u do 4^(log(base 8) of 27 without a calculator\r
\n" ); document.write( "\n" ); document.write( "Problem: 4^(log(27)/log(8)) \n" ); document.write( "
Algebra.Com's Answer #165875 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
Edwin's full explanation:
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document.write( "\"4%5E%28log%288%2C27%29%29\"\r\n" );
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document.write( "Since there is a \"4\" and an \"8\" involved, and\r\n" );
document.write( "both are powers of \"2\", let's try to get as much\r\n" );
document.write( "as possible involving the number \"2\":\r\n" );
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document.write( "Let's first work only on the exponent\r\n" );
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document.write( "\"log%288%2C27%29\"\r\n" );
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document.write( "Use the change of base formula \r\n" );
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document.write( "to change base \"8\" to base \"2\"\r\n" );
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document.write( "\"log%282%2C27%29%2Flog%282%2C8%29\"\r\n" );
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document.write( "Write \"27\" as \"3%5E3\" and \"8\" as \"2%5E3\"\r\n" );
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document.write( "\"log%282%2C3%5E3%29%2Flog%282%2C2%5E3%29\"\r\n" );
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document.write( "Use the rule \"log%28%28B%29%2CX%5EN%29=N%2Alog%28%28B%29%2CX%29\" to rewrite\r\n" );
document.write( "numerator and denominator:\r\n" );
document.write( "\r\n" );
document.write( "\"%283%2Alog%282%2C3%29%29%2F%283%2Alog%282%2C2%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Cancel the \"3\"'s:\r\n" );
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document.write( "\"%28cross%283%29%2Alog%282%2C3%29%29%2F%28cross%283%29%2Alog%282%2C2%29%29\"\r\n" );
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document.write( "and we have just:\r\n" );
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document.write( "\"%28log%282%2C3%29%29%2F%28log%282%2C2%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Use the rule \"log%28B%2C%28B%29%29=1\" to rewrite the denominator:\r\n" );
document.write( "\r\n" );
document.write( "\"%28log%282%2C3%29%29%2F1\"\r\n" );
document.write( "\r\n" );
document.write( "or just\r\n" );
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document.write( "\"log%282%2C3%29\"\r\n" );
document.write( "\r\n" );
document.write( "Now let's go back to the original problem, which was:\r\n" );
document.write( "\r\n" );
document.write( "\"4%5E%28log%288%2C27%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "and replace the exponent with \"log%282%2C3%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"4%5Elog%282%2C3%29\"\r\n" );
document.write( "\r\n" );
document.write( "Rewrite the base \"4\" as \"%282%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"%282%5E2%29%5Elog%282%2C3%29\"\r\n" );
document.write( "\r\n" );
document.write( "Use the rule \"%28B%5EM%29%5EN=B%5E%28M%2AN%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"2%5E%282%2Alog%282%2C3%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Use the rule \"N%2Alog%28%28B%29%2CX%29=log%28%28B%29%2CX%5EN%29\" to rewrite the exponent:\r\n" );
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document.write( "\"2%5E%28log%282%2C3%5E2%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Write \"3%5E2\" as \"9\"\r\n" );
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document.write( "\"2%5E%28log%282%2C9%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Use the rule \"log%28B%2C%28B%5EN%29%29+=+N\"\r\n" );
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document.write( "Final answer: \r\n" );
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document.write( "\"9\"\r\n" );
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document.write( "--------------------------------\r\n" );
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document.write( "Again we notice there is a \"4\" and an \"8\" involved,\r\n" );
document.write( "and both are powers of \"2\", let's try to get as much\r\n" );
document.write( "as possible involving the number \"2\":\r\n" );
document.write( "\r\n" );
document.write( "As before we first work only with the exponent:\r\n" );
document.write( "\r\n" );
document.write( "\"log%2827%29%2Flog%288%29\"\r\n" );
document.write( "\r\n" );
document.write( "We write \"27\" as \"3%5E3\" and \"8\" as \"2%5E3\"\r\n" );
document.write( "\r\n" );
document.write( "\"log%283%5E3%29%2Flog%282%5E3%29\"\r\n" );
document.write( "\r\n" );
document.write( "Use the rule \"log%28%28B%29%2CX%5EN%29=N%2Alog%28%28B%29%2CX%29\" to rewrite\r\n" );
document.write( "numerator and denominator:\r\n" );
document.write( "\r\n" );
document.write( "\"%283%2Alog%283%29%29%2F%283%2Alog%282%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Cancel the \"s\"'s\r\n" );
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document.write( "\"%28cross%283%29%2Alog%283%29%29%2F%28cross%283%29%2Alog%282%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"log%283%29%2Flog%282%29\"\r\n" );
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document.write( "When no base is written, the base is assumed to be \"10\"\r\n" );
document.write( "\r\n" );
document.write( "\"log%2810%2C3%29%2Flog%2810%2C2%29\"\r\n" );
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document.write( "Write that fraction as a division:\r\n" );
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document.write( "\"log%2810%2C3%29\"\"%22%F7%22\"\"log%2810%2C2%29\"\r\n" );
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document.write( "Use the change of base formula:  on each to get them\r\n" );
document.write( "to logs base 2.\r\n" );
document.write( "\r\n" );
document.write( "\"log%282%2C3%29%2Flog%282%2C10%29\"\"%22%F7%22\"\"log%282%2C2%29%2Flog%282%2C10%29\"\r\n" );
document.write( "\r\n" );
document.write( "Use the rule \"log%28B%2C%28B%29%29+=+1\" to rewrite \"log%282%2C2%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"log%282%2C3%29%2Flog%282%2C10%29\"\"%22%F7%22\"\"1%2Flog%282%2C10%29\"\r\n" );
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document.write( "Invert the second fraction and change division to multiplication:\r\n" );
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document.write( "\"log%282%2C3%29%2Flog%282%2C10%29\"\"%22%D7%22\"\"log%282%2C10%29%2F1\"\r\n" );
document.write( "\r\n" );
document.write( "Cancel:\r\n" );
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document.write( "\"log%282%2C3%29%2Fcross%28log%282%2C10%29%29\"\"%22%D7%22\"\"cross%28log%282%2C10%29%29%2F1\"\r\n" );
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document.write( "All that's left is \r\n" );
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document.write( "\"log%282%2C3%29\"\r\n" );
document.write( "\r\n" );
document.write( "Now let's go back to the original problem, which was:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and replace the exponent by the simpler \"log%282%2C3%29\"\r\n" );
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document.write( "\"4%5Elog%282%2C3%29\" \r\n" );
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document.write( "Wow!  That just happens by chance to be the exact same\r\n" );
document.write( "thing as we got in the first problem, so we just copy\r\n" );
document.write( "the above from there on and, of course, it will have\r\n" );
document.write( "the same final answer.\r\n" );
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document.write( "Rewrite the base \"4\" as \"%282%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"%282%5E2%29%5Elog%282%2C3%29\"\r\n" );
document.write( "\r\n" );
document.write( "Use the rule \"%28B%5EM%29%5EN=B%5E%28M%2AN%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"2%5E%282%2Alog%282%2C3%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Use the rule \"N%2Alog%28%28B%29%2CX%29=log%28%28B%29%2CX%5EN%29\" to rewrite the exponent:\r\n" );
document.write( "\r\n" );
document.write( "\"2%5E%28log%282%2C3%5E2%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Write \"3%5E2\" as \"9\"\r\n" );
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document.write( "\"2%5E%28log%282%2C9%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Use the rule \"log%28B%2C%28B%5EN%29%29+=+N\"\r\n" );
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document.write( "Final answer: \r\n" );
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document.write( "\"9\" \r\n" );
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document.write( "Edwin
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