Transcription of 3.3 The logarithm as an inverse function
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3.3 The logarithm as an inverse function
The logarithm as an inverse functionIn this section we concentrate onunderstandingthe logarithm function . If the logarithm is understoodas the inverse of the exponential function , then the variety of properties of logarithms will be seen asnaturally flowing out of our rules for The meaning of the logarithmThe logarithmic functiong(x) = logb(x) is the inverse of an exponential functionf(x) = so themeaningofy= logb(x) isby= expressionby=xis said to be the exponential form for thelogarithmy= logb(x). The positive constantbis called thebase(of the logarithm .)Some worked each of the following logarithms in exponential form and then use that exponential form tosolve log2(8) = exponential form is 2x= 23= 8 the answer isx= log2(247) = exponential form is 2x= log2(12) = exponential form is 2x= 2 1=12the answer isx= log2(18) = exponential form is 2x= 2 3=18the answer isx= log2(3 2) = exponential form is 2x= 21 1 The graph of a logarithm functionThe graph ofy= 2xwas drawn in an earlier lectur
3.To nd the inverse of y= 5 + ex we swap variables, subtract 5 from both sides and then take the natural log to get ln(x 5) = y:So f 1(x) = ln(x 5) : 4.To nd the inverse of f(x) = log 2 (x+ 2) + 2 we write y= log 2 (x+ 2) + 2; change variables (to indicate that we are swapping inputs and outputs)
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