S&Z 6.2 & 6.3 - 6.2 PROPERTIES OF LOGARITHMS 6.2 PROPERTIES...

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6.2 PROPERTIESOFLOGARITHMS16.2 PROPERTIESOFLOGARITHMSIn Section 6.1, we introduced the logarithmic functions as inverses of exponential functions and discussed afew of their functional properties from that perspective. In this section, we explore the algebraicproperties of logarithms. Historically, these have played a huge role in the scientific developmentof our society since, among other things, they were used to develop analog computing devicescalled slide rules which enabled scientists and engineers to perform accurate calculations leadingto such things as space travel and the moon landing. As we shall see shortly, logs inherit analogsof all of the properties of exponents you learned in Elementary and Intermediate Algebra. We firstextract two properties from Theorem 6.2 to remind us of the definition of a logarithm as theinverse of an exponential function.Theorem 6.3. (Inverse Properties of Exponential and Log Functions) Let b >0, 1.ba= c if and only if logb(c) = logb(bx)= x for all xand blogb(x)= xfor all x >Next, we spell out what it means for exponential and logarithmic functions to be one-to-one.Theorem 6.4. (One-to-one Properties of Exponential and Log Functions)bxand g(x)= logb(x) where b > 0, b = 1. Then f and gare one-to-one. In other words:bu= bwif and only if u = w for all real numbers u and w.logb(u) = logb(w) if and only if u = w for all real numbers u > 0, w > 0We now state the algebraic properties of exponential functions which will serve as a basis for the properties oflogarithms. While these properties may look identical to the ones you learned in Elementary andIntermediate Algebra, they apply to real number exponents, not just rational exponents. Note thatin the theorem that follows, we are interested in the properties of exponential functions, so thebase b is restricted to b > 0, b = 1. An added benefit of this restriction is that it eliminates thepathologies discussed in Section 5.3 when, for example, we simplified (x2/3)andobtained |x| instead of what we had expected from the arithmetic in the exponents, x1= x.Theorem 6.5. (Algebraic Properties of Exponential Functions) Let f(x) = bxbe anexponential function (b > 0, b = 1) and let u and w be real numbers.Product Rule: f(u + w) = f(u)f(w). In other words, bu+w= bubf(u)buQuotient Rule: f (u — w) =.In other words, bu-w=Jf(w)bwPower Rule: (f (u))w= f (uw). In other words, (bu)w= buwWhile the properties listed in Theorem 6.5 are certainly believable based on similar properties of integer andrational exponents, the full proofs require Calculus. To each of these properties ofb= a0.w^
2EXPONENTIALANDLOGARITHMICFUNCTIONSexponential functions corresponds an analogous property of logarithmic functions. We list these below in ournext theorem.There are a couple of different ways to understand why Theorem 6.6is true. Consider the product rule: logb(uw)= logb(u) + logb(w). Let a= logb(uw), c = logb(u), and d = logb(w). Then, by definition, ba= uw,= u and bd= w. Hence, ba= uw= bcbd= bc+d, so that ba= bc+d. By the one-to-one property ofbx,we have a= c + d. In other words, log
bc

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