document.write( "Question 974132: explain why log(ab)= log(a) + log(b) \n" ); document.write( "
Algebra.Com's Answer #596135 by Edwin McCravy(20056)\"\" \"About 
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log(ab)= loga + logb
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document.write( "First a little background.  Then the proof:\r\n" );
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document.write( "It comes from the fact that to multiply powers of the same base, you simply\r\n" );
document.write( "add exponents.\r\n" );
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document.write( "log(a number) is the power to which you must raise 10 to get that number. \r\n" );
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document.write( "The exponent to which you must raise 10 to get ab is the sum of the power to\r\n" );
document.write( "which you must raise 10 to get \"a\" and the power to which you must raise 10 to\r\n" );
document.write( "get b.\r\n" );
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document.write( "Here's the proof:\r\n" );
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document.write( "Let log(ab) = p, log(a) = q, and log(b) = r.\r\n" );
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document.write( "Then what we want to prove is that p = q+r\r\n" );
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document.write( "Those three logarithm equations by definition of logarithms are equivalent\r\n" );
document.write( "respectively to these three exponential equations:\r\n" );
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document.write( "10p = ab, 10q = a, 10r = b\r\n" );
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document.write( "Multiply the last two equation (multiply equals by equals):\r\n" );
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document.write( "10q10r = ab\r\n" );
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document.write( "Multiply on the left by adding exponents of 10:\r\n" );
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document.write( "10q+r = ab\r\n" );
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document.write( "But also 10p = ab, so\r\n" );
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document.write( "10p = 10q+r\r\n" );
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document.write( "The bases are the same positive number other than 1, 10.\r\n" );
document.write( "Therefore the exponents must be equal:\r\n" );
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document.write( "p = q+r\r\n" );
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document.write( "log(ab)= log(a) + log(b)\r\n" );
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document.write( "Edwin

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