document.write( "Question 87657: If loga(m) = p and logb(m) = q (a&b are bases)
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Algebra.Com's Answer #63573 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "If loga(m) = p and logb(m) = q (a&b are bases) \r\n" );
document.write( "prove that \"%28p-q%29%2F%28p%2Bq%29\" = \"%28log%28%28b%29%29-log%28%28a%29%29%29%2F%28log%28%28b%29%29%2Blog%28%28a%29%29%29\"\r\n" );
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document.write( " loga(m) = p   and   logb(m) = q\r\n" );
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document.write( "Rewrite these log equations as exponential equations:\r\n" );
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document.write( "      ap = m              bq = m \r\n" );
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document.write( "Now take the log10 of both sides.  When the\r\n" );
document.write( "base is 10 we do not write the base, as it is understood\r\n" );
document.write( "to be 10 when it is not written:\r\n" );
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document.write( " log(ap) = log(m)    log(bp) = log(m)\r\n" );
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document.write( "Use the rule of logs that allows you to move the exponent\r\n" );
document.write( "of what the log is taken of out in friont as a multiplier:\r\n" );
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document.write( "p·log(a) = log(m)      q·log(b) = log(m) \r\n" );
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document.write( "Solve for p and q:\r\n" );
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document.write( "     log(m)                        log(m)\r\n" );
document.write( "p = --------                  q = --------\r\n" );
document.write( "     log(a)                        log(b)\r\n" );
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document.write( "Let's find p - q\r\n" );
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document.write( "         log(m)     log(m)\r\n" );
document.write( "p - q = -------- - --------  \r\n" );
document.write( "         log(a)     log(b)\r\n" );
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document.write( "Get LCD = log(a)log(b)\r\n" );
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document.write( "         log(m)log(b)     log(m)log(a)\r\n" );
document.write( "p - q = -------------- - -------------- \r\n" );
document.write( "         log(a)log(b)     log(b)log(a)\r\n" );
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document.write( "         log(m)log(b) - log(m)log(a)\r\n" );
document.write( "p - q = ---------------------------- \r\n" );
document.write( "               log(a)log(b)\r\n" );
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document.write( "=====================================\r\n" );
document.write( "Similarly, let's find P + q\r\n" );
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document.write( "         log(m)     log(m)\r\n" );
document.write( "p + q = -------- + --------  \r\n" );
document.write( "         log(a)     log(b)\r\n" );
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document.write( "         log(m)log(b)     log(m)log(a)\r\n" );
document.write( "p + q = -------------- + -------------- \r\n" );
document.write( "         log(a)log(b)     log(b)log(a)\r\n" );
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document.write( "         log(m)log(b) + (log m)(log a)\r\n" );
document.write( "p + q = -------------------------------- \r\n" );
document.write( "               log(a)log(b)   \r\n" );
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document.write( "==========================================\r\n" );
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document.write( "Now divide p - q by p + q\r\n" );
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document.write( " p - q     log(m)log(b) - log(m)log(a)     log(m)log(b) + log(m)log(a)\r\n" );
document.write( "------- = ----------------------------- ÷ -----------------------------\r\n" );
document.write( " p + q          log(a)log(b)                      log(a)log(b)\r\n" );
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document.write( "Invert the second fraction and multiply\r\n" );
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document.write( " p - q     log(m)log(b) - log(m)log(a)           log(a)log(b)\r\n" );
document.write( "------- = ----------------------------- × -----------------------------\r\n" );
document.write( " p + q         log(a)log(b)                log(m)log(b) + log(m)log(a)\r\n" );
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document.write( "Cancel the (log a)(log b)'s\r\n" );
document.write( "                                                        1\r\n" );
document.write( " p - q     log(m)log(b) - log(m)log(a)           log(a)log(b)\r\n" );
document.write( "------- = ----------------------------- × -----------------------------\r\n" );
document.write( " p + q         log(a)log(b)                log(m)log(b) + log(m)log(a)\r\n" );
document.write( "                      1\r\n" );
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document.write( " p - q     log(m)log(b) - log(m)log(a)        \r\n" );
document.write( "------- = ----------------------------- \r\n" );
document.write( " p + q     log(m)log(b) + log(a)log(a)\r\n" );
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document.write( "Factor log(m) out of top and bottom on the right side:\r\n" );
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document.write( " p - q     log(m)[log(b) - log(a)]        \r\n" );
document.write( "------- = ------------------------- \r\n" );
document.write( " p + q     log(m)[log(b) + log(a)]\r\n" );
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document.write( "Cancel the (log m)'s\r\n" );
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document.write( "             1\r\n" );
document.write( " p - q     log(m)[log(b) - log(a)]        \r\n" );
document.write( "------- = ------------------------- \r\n" );
document.write( " p + q     log(m)[log(b) + log(a)]\r\n" );
document.write( "              1 \r\n" );
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document.write( "              \r\n" );
document.write( " p - q     log(b) - log(a)        \r\n" );
document.write( "------- = ----------------- \r\n" );
document.write( " p + q     log(b) + log(a)\r\n" );
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document.write( "Edwin
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