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
and solve for yby taking square roots of both sides p lnx= y: So one inverse function is f 1(x) = p lnx: 2.To nd the inverse of y= ex2 5 we swap letters so that x= ey2=5, take natural logs of both sides lnx= y2 5; add 5 and take square roots so that f 1(x) = p lnx+ 5 : 124
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