121616949-math.87 - 4.6 Exponential and Logarithmic...

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4.6 Exponential and Logarithmic functions 73 Now let’s make this a bit more general. Suppose A and B are two numbers, and A = a x , B = a y . Then log a ( AB ) = log a ( a x a y ) = log a ( a x + y ) = x + y = log a ( A ) + log a ( B ). Now consider ( a 5 ) 3 = a 5 · a 5 · a 5 = a 5+5+5 = a 5 · 3 = a 15 . Again it’s clear that more generally ( a m ) n = a mn , and again this gives us a fact about logarithms. If A = a x then A y = ( a x ) y = a xy , so log a ( A y ) = xy = y log a ( A )—the exponent can be “pulled out in front.” We have cheated a bit in the previous two paragraphs. It is obvious that a 5 = a · a · a · a · a and a 3 = a · a · a and that the rest of the example follows; likewise for the second example. But when we consider an exponential function a x we can’t be limited to substituting integers for x . What does a 2 . 5 or a - 1 . 3 or a π mean? And is it really true that a 2 . 5 a - 1 . 3 = a 2 . 5 - 1 . 3 ? The answer to the first question is actually quite difficult, so we will evade it; the answer to the second question is “yes.”
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