Math M119 Ch 1 Sec 6 Notes
Math M119 Ch 1 Sec 6 Notes M119
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Date Created: 01/25/16
Chapter 1 Section 6 The Logarithm Function Logarithmic functions go handinhand with exponential functions Logarithmic functions are represented by expressions such as the following 10g10x 10g2x 10g37x Alogex In each of these expressions the subscript is referred to as the base of the logarithm The base will always be a positive number Although any positive number can be used as the base as explained below we ll restrict ourselves to values for the base that are greater than 1 The most commonly used bases are 10 2 and e We read expressions such as those above by using phrases such as quotThe base 10 logarithm of xquot or Log base 2 of xquot What do these expressions above mean Take the expression 10g37x for example The symbolism logelx represents a number Speci cally it represents the exponent that must be placed on 37 to produce the number x Examples log10 10000 4 Why Because 10 10000 log25129 Why Because 292512 10g64413 Why Because641324 log37 19 08154267 Why Because 3708154267quot 19 Remember that a function is characterized as a relationship that takes an input value and associates with it a unique output value These qualities are present in the logarithm If a base bgt1 is given and some positive number xgt0 has been speci ed then there will be a unique real number y for which byzx From this quality a function is created flxllogblxl If a base bgt1 is given then the function flxllogblxl has as its domain the set of all positive real numbers and has as its range the set of all real numbers To express this in 39 R mathematical notation we sometimes Just write 13 The base b must be a positive real number If you think about alternatives to this you quickly understand why A base of 1220 would be an idea of little interest since 0y0 always for every nonzero number y A negative number base blt0 would be an idea of little interest as well for consider the example of trying to work with a base of 2 You could work out a value for log216l That value would be 4 since 2416 But you would encounter a roadblock trying to work out a value for 10328 You won t nd an exponent to use on 2 that will produce an 8 log2x is unde nable for most every number x This is why the base b must be a positive real number Beyond these observations if we consider a base of 1 then this too seems of little value since log1l5 for example will be unde ned owing to the fact that no matter what exponent is placed on 1 you will never produce a 5 as a result Although having a base that is a number between O and 1 makes good sense by convention we restrict the base of logarithms to the set of real numbers that are greater than The independent variable the input values for the function must be a positive real number The reason for this is simple If you place any exponent on a positive real number b then you will produce a positive result For example 10g1o100 is not de nable since there are no exponents possible that could be used with 10 that would produce 100 as a result There is one base that tends to be favored over all of the other possibilities It is the base that is represented by the letter e e271828 where the trailing dots are meant to indicate that the digits continue on endlessly and show no repeating pattern of the type one nds in the decimal representation of a rational number The reason that the number e is favored as a base for logarithms will become clear once we begin to develop some of the calculus ideas about the derivative in the coming weeks Our work with theses calculus ideas will show that when a base of e is used then calculus problems that involve logarithms tend to work out a little more simply than they do when other logarithm bases are used For this reason we refer to the logarithm base e as the Natural Logarithm When a logarithm is using a base of e we also simplify the notation from logeix to the simpler lnix The following Identities provide useful Properties of logarithms blogbxx This is easy to see just by thinking of the de nition of logarithms 10grsubb le ight is by de nition the exponent that must be placed on the base b to produce the result x logbbxx This is easy to see just by thinking of the de nition of logarithms logmub 3 16W A High is by de nition 6 the exponent that must be placed on the base b to produce the result bx That exponent is obviously the number x 10gbxy10gbx10gby This is easy to see just by thinking of the properties of exponents along with the logarithm property mentioned in the previous box logblxy represents the unique exponent that must be used on the base b to produce the result of xy Notice that bIOgbl CtIOgbly191 ngbIOgblyExy Therefore the expression logblxllogblyl also represents that same unique exponent that must be used on the base b to produce the result of xy Thetwoexpressionslogbxylogbxlogby eaCh FEDFESent the same number IV 10gbXy10gbx10gby The same form of reasoning that was used to explain the previous identities can be used to explain this one tOO V If xgt1 then logbxgt0 10gb1O If 0ltxlt1 then This is easy to see just by thinking of the de nition of logarithms 10gbxlt0 V logbxquotnlogbx To see that this is true use property I to note the following blogbx n X bn1ogbxltblogbxgtnltxgtn xn Hence both of the expressions logblx Anlogblxl each represent the unique exponent that can be used on b to produce the value xquot and so both of these expressions represent the same number With the help of logarithms it is easy to rewrite an exponential function to have any desired base Example Let fx7315x Rewrite fix as an exponential function having a base of 10 We want 7315xP010r It is easy to see that Po will need to be 7 That just leave us trying to determine r so that 315x10r Here we can use the identity 315101 g1 l33915l That means 315x101 g10l33915lx10x1 g10l315l This makes clear that the needed exponent on 10 will be rx10g10315 and we can now write fx710xlog10315z71004983x As examples work the following problems from page 50 of our text book 2 16 26 30 34 38 39
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