CHAPTER 5 Eponential and Logarithmic Functions We have eamined power functions like f =. Interchanging and ields a different function f =. This new function is radicall different from a power function and has vastl different properties. It is called an eponential function. Eponential functions have man applications and pla a big role in this course. Working with them requires understanding the basic laws of eponents. This chapter reviews these laws before recalling eponential functions. Then it eplores inverses of eponential functions, which are called logarithms. Recall that in an epression such as a n in which a is raised to the power of n, the number a is called the base and n is the eponent. 5. Review of Eponents We start at the beginning. For a number a and a positive integer n, a n = a a {{ a a. n times For eample, 5 = = 3. This would be too elementar to mention ecept that ever significant eponent propert flows from it. For eample, ab n = ab ab ab ab {{ n times = a a {{ a a n times b b b b {{ n times = a n b n, that is, ab n = a n b n. Also, a n b = a b a b a b = an b. And a m a n = a m+n because n a m a n = a a {{ a a a a {{ a a = a m+n. m times n times Assuming for the moment that m > n, we also get a m a n = m times {{ a a a a a a a a a a {{ n times = a a {{ a = a m n m n times
60 Eponential and Logarithmic Functions because the a s on the bottom cancel with n of the a s on top, leaving m n a s on top. Also notice that a n m = a nm because m times {{ a n m = a a {{ a a a a {{ a a a a {{ a a = a nm. n times n times n times We have just verified the following fundamental Laws of eponents. Basic Laws of Eponents a = a ab n = a n b n a b a m a n = a m+n a m a n = am n n = a n b n a n m = a mn So far we have assumed n is a positive integer because in performing a n = a a a we cannot multipl a times itself a negative or fractional number of times. But there is a wa to understand these rules when n is zero, negative or fractional. Trusting the above propert a m n = am a ields n a 0 = a = a a =, provided a 0 a n = a 0 n = a0 a n = a n. Notice 0 0 is undefined because above we have 0 0 = 0 = 0 0 0, which is undefined. But we can find a n when n is 0 and a 0 or negative, as in 3 = = 3 8. In essence we just multiplied times itself 3 times! Also note a = a. What about fractional powers, like a /n? If we believe a n m = a nm, then a n n = a n n = a = a. Thus a /n n = a, meaning a /n = n a. So 6 /4 = 4 6 = and / =. Further, a m n = a n m = a m n = n a m. In summar: a 0 = if a 0 a n = a n a = a a n = n a a m n = n a m = n a m The boed equations hold for an rational values of m and n, positive or negative. We will use these frequentl, and without further comment.
Review of Eponents 6 Eample 5. Knowing the rules of eponents in the boes above means we can evaluate man epressions without a calculator. For eample, suppose we are confronted with 6.5. What number is this? We reckon 6.5 = 6 3/ = 6 3/ = 6 3 = 4 3 = 64. For another eample, 8.5 = 8 3/ = 8 3/ = 3 = 8 = 3 3 3 = 6. Also, 8 5/3 = 3 8 5 = 5 = = 3. But if we attempt 8 5/, we run into a problem because 8 5/ = 8 5, and 8 is not defined or at least it is not a real number. In this case we simpl sa that 8 5/ is not defined. It is important to be able to work problems such as these fluentl, b hand or in our head. Over-reliance on calculators leads weak algebra skills, which will defeat ou later in the course. If our algebra is rust it is good practice to write everthing out, without skipping steps. Algebra skills grow quickl through usage. We have now seen how to evaluate a p provided that p is a positive or negative integer or rational number i.e., fraction of two integers. But not ever power p falls into this categor. For eample, if p = π, then it is impossible to write p = m n, as a fraction of two integers. How would we make sense of something like π? One approach involves approimations. As π 3.4, we have π 3.4 = 34 00 00 = 34. We could, at least in theor, arrive at ver good approimations of π with better approimations of π. We will drop this issue for now, because we will find a perfect wa to understand a for an once we have developed the necessar calculus. For now ou should be content knowing that a has a value for an, provided that a is a positive number. Eercises for Section 5. Work the following eponents with pencil and paper alone. Then compare our answer to a calculator s to verif that the calculator is working properl.. 5 /. 4 / / 3. 4 4. 7 /3 5. 7 /3 6. 7 /3 7. 7 4/3 8. 9. 0. 3. 5. 6 6. 4 9 / 4 00 7. 3 8. 3 94 3 9.. 3 3 0. 3 3. 3 9 3 7 3 3. / 4. 4
6 Eponential and Logarithmic Functions 5. Eponential Functions An eponential function is one of form f = a, where a is a positive constant, called the base of the eponential function. For eample f = and f = 3 are eponential functions, as is f =. If we let a = in f = a we get f = =, which is, in fact, a linear function. For this reason we agree that the base of an eponential function is never. To repeat, an eponential function has form f = a, where a is a positive constant unequal to. We require a to be positive because we saw that in Eample 5. that a ma not be defined if a is negative. Let s graph the eponential function f =. Below is a table with some 8 f = sample and f values. The resulting 7 graph is on the right. 6 3 0 3 f = 8 4 4 8 3 Notice that f = is positive for an, but gets closer to zero the as moves in the negative direction. But as > 0 for an, the graph never touches the -ais. 3 3 The function f = grows ver rapidl as moves in the positive direction. Because f = is defined for all real numbers the domain of f is R. The graph suggests that the range is all positive real numbers, that is, the range is the interval 0,. Let s look at the eponential function f =. Here is its table and graph. 3 0 3 f = 8 4 Unlike =, this function decreases as increases. In fact, f = = =. As increases towards infinit the denominator becomes ever bigger, so the fraction f = gets closer and closer to zero. But no matter how big gets, we still have f = > 0. 4 8 f = 5 4 8 3 3 7 6 5 4 3
Eponential Functions 63 Figure 5. shows some additional eponential functions. It underscores the fact that the domain of an eponential function is R =,. The range is 0,. The -intercept of an eponential function is. 0 f = 0 f = 3 f = f =.5 3 0 f =.5 f = f = Figure 5.. Some eponential functions. Each eponential function f = a has -intercept f 0 = a 0 =. If a >, then f = a increases as increases. If 0 < a <, then f = a decreases as increases. Eercises for Section 5.. Add the graphs of eponential functions f = 0 and f = 3 to Figure 5... Add graphs of f =.5 = 0. 6 and f =.5 = 0.8 to Figure 5.. 3. Use graph shifting techniques from Section.4 to draw the graph of = 3 +. 4. Use graph shifting techniques from Section.4 to draw the graph of = +. 5. Its graph shows that the eponential function f = is one-to-one. Use ideas from Section 4. to draw the graph of f. 6. Consider the inverse of f =. Find f 8, f 4, f, f, and f.
64 Eponential and Logarithmic Functions 5.3 Logarithmic Functions Now we appl the ideas of Chapter 4 to eplore inverses of eponential functions. Such inverses are called logarithmic functions, or just logarithms. f = a An eponential function f = a is one-to-one and thus has an inverse. As illustrated above, this inverse sends an number to the number for which f =, that is, for which a =. In other words, f = the number for which a = From this it seems that a better name for f might be a, for then a = the number for which a = The idea is that a is the number that goes in the bo so that a =. Using a as the name of f thus puts the meaning of f into its name. We therefore will use the smbol a instead of f for the inverse of f = a. For eample, the inverse of f = is a function called, where = the number for which = Consider 8. Putting 3 in the bo gives 3 = 8, so 8 = 3. Similarl 6 = 4 because 4 = 6, 4 = because = 4, = because =, 0.5 = because = = 0.5....
Logarithmic Functions 65 In the same spirit the inverse of f = 0 is a function called 0, and Therefore we have 0 = the number for which 0 = 0 000 = 3 because 0 3 = 000, 0 0 = because 0 = 0, 0 0. = because 0 = 0 = 0.. Given a power 0 p of 0 we have 0 0 p = p. For eample, 0 00 = 0 0 =, 0 0 = 0 0 / =. But doing, sa, 0 5 is not so eas because 5 is not an obvious power of 0. We will revisit this at the end of the section. In general, the inverse of f = a is a function called a, pronounced a bo, and defined as a = the number for which a = You can alwas compute a of a power of a in our head because a a p = p. The notation a is nice because it reminds us of the meaning of the function. But this book is probabl the onl place that ou will ever see the smbol a. Ever other tetbook in fact all of the civilized world uses the smbol log a instead of a, and calls it the logarithm to base a... Definition 5. the function Suppose a > 0 and a. The logarithm to base a is log a = a = number for which a =. The function log a is pronounced log base a. It is the inverse of f = a. Here are some eamples. log 8 = 8 = 3 log 5 5 = 5 5 = 3 log 4 = 4 = log 5 5 = 5 5 = log = = log 5 5 = 5 5 = log = = 0 log 5 = 5 = 0
66 Eponential and Logarithmic Functions To repeat, log a and a are different names for the same function. We will bow to convention and use log a, mostl. But we will revert to a whenever it makes the discussion clearer. Understanding the graphs of logarithm functions is important. Because log a is the inverse of f = a, its graph is the graph of = a reflected across the line =, as illustrated in Figure 5.. = f = a = ln 7 6 5 4 3 3 3 4 5 6 7 = f = a = log a Figure 5.. The eponential function = a and its inverse = log a. Notice log a is negative if 0 < <, Also log a tends to as gets closer to 0. Take note that the domain of log a is all positive numbers 0, because this is the range of a. Likewise the range of log a is the domain of a, which is R. Also, because log a = a = 0, the -intercept of = log a is. The logarithm function log 0 to base 0 occurs frequentl enough that it is abbreviated as log and called the common logarithm. Definition 5. The common logarithm is the function log defined as log = log 0 = 0. Most calculators have a log button for the common logarithm. Test this on our calculator b confirming that log000 = 3 and log0. =. The button will also tell ou that log5 =.76095. In other words 0 5 =.76095, which means 0.76095 = 5, a fact with which our calculator will concur. One final point. Convention allows for log a in the place of log a, that is, the parentheses ma be dropped. We will tend to use them.
Logarithm Laws 67 Eercises for Section 5.3. log 3 8 =. log 3 9 = 3. log3 3 = 4. log 3 3 = 5. log 3 = 6. log000 = 7. log 3 0 = 8. log 3 00 = 9. log0.0 = 0. log =. log 4 4 =. log 4 = 3. log 4 = 4. log 4 6 = 5. log 4 8 = 6. Simplif log sin. 7. Simplif 0 log5. 8. Draw the graphs of = 0 and its inverse = log. 9. Draw the graphs of = 3 and its inverse = log 3. 5.4 Logarithm Laws The function log a, also called a, is the inverse of the eponential function f = a. Consequentl as this section will show the laws of eponents outlined in Section 5. come through as corresponding laws of log a. To start, for an it is obvious that a a = because is what must go into the bo so that a to that power equals a. So we have a a =, log a a =. 5. This simpl reflects the fact that f f = for the function f = a. Net consider the epression a a. Here a is being raised to the power a, which is literall the power a must be raised to to give. Therefore a a =, a log b = 5. for an in the domain of a. This is just saing f f =. The in Equations 5. and 5. can be an appropriate quantit or epression. It is reasonable to think of these equations as saing a a = and a a =, where the gra rectangle can represent an arbitrar epression. eamples, a a + +3 = + + 3 and a a sinθ = sinθ. For
68 Eponential and Logarithmic Functions Now we will verif a ver fundamental formula for log a. Notice a = a a a a a using = a a and = a a = a a a +a because a c a d = a c+d = a + a using a a = We have therefore established a = a + a, log a = log a + log a. 5.3 B the same reasoning ou can also show a = a a, that is, a log a = a a, = log a log a. 5.4 Appling a = 0 to this ields a = a a = a, so a log a = a, = log a. 5.5 Here is a summar of what we have established so far. Logarithm Laws log a a = log a = 0 log a = log a + log a log a = log a a log a = log a a = log a = log a log a log a = log a The one law in this list that we have not et verified is log a = log a. This is an especiall useful rule because it sas that taking log a of converts the eponent to a product. Because products tend to be simpler than eponents, this propert is tremendousl useful in man situations. To verif it, just notice that a = a a a because = a a = a a a because a b = a b = a using a a =
Logarithm Laws 69 Therefore a = a, or log a = log a, as listed above. B the above laws, certain epressions involving logarithms can be transformed into simpler epressions. Eample 5. Write log 5 + log + log 5 as an epression with a single logarithm. To solve this we use the laws of logarithms, and work as follows. log 5 + log + log 5 = log 5 + log + / log 5 = log 5 + log + log 5 = log 5 + log 5 5 + = log 5 + = log Eample 5.3 Break the epression log sin 0 into simpler logarithms. Recall that log is the common logarithm log 0. Using the laws of logarithms, sin sin / log 0 = log 0 = sin log 0 = logsin log0 = logsin = log + logsin = log + log sin. The final answer is somewhat subjective. What do we mean b simpler logarithms? An alternate answer would be log + log sin.
70 Eponential and Logarithmic Functions Eample 5.4 Simplif log 8 log 7. To solve this we use the laws of logarithms to get 8 log 8 log 7 = log 7 = log 4 =. Now work a few practice problems. Eercises for Section 5.4. Write as a single logarithm: 5log 3 + + log log 3. Write as a single logarithm: log sin + log 4 3log 3 3. Write as a single logarithm: + log5 + log7 4. Write as a single logarithm: log + log5 5. Break into simpler logarithms: 3 log + 6. Break into simpler logarithms: log + 5 4 7 cos 7. Break into simpler logarithms: log + 3 6 3 8. Break into simpler logarithms: log 3 5 3 9. Simplif: log 0 0 0. Simplif: log + log5. Simplif: log + log + log5. Simplif: log log 5 + log 0 3. Find the inverse of the function f = 7 3 +4 4. Find the inverse of the function f = 4log + 3 5. Find the inverse of the function gθ = 5 θ + 6. Find the inverse of the function gz = /z 7. Use graph shifting to sketch the graph of = log +. 8. Use graph shifting to sketch the graph of = log + 4.
The Natural Eponential and Logarithm Functions 7 5.5 The Natural Eponential and Logarithm Functions Calculus involves two ver significant constants. One of them is π, which is essential to the trigonometric functions. The other is a msterious number e =.78889459 that arises from eponential functions. Like π, the number e is irrational it is not a fraction of integers and its digits do not repeat. But unlike π the ratio of circumference to diameter of a circle it is difficult to justif the importance of e without first developing some calculus. This will come in due time. Eventuall we will find formulas for e. For now we merel remark that e will turn out to be ver significant. The eponential function e is called the natural eponential function. The natural eponential function is the function f = e. The natural eponential function = e is graphed on the right. Because < e < 3, this graph lies between the graphs of the eponential functions = and = 3, whose graphs are sketched in gra. 8 7 6 5 = The natural eponential function 4 shares features of other eponential 3 functions. It grows ver rapidl as moves in the positive direction. Taking in the negative direction, e becomes closer and closer to 0. Its domain of f = e is R and the range is 0,. 3 3 Later we will for the most part forsake all other eponential functions and treat the natural eponential function as if it is the onl eponential function. There are good reasons for doing this, as we will see. Section 5.3 tells us the inverse of f = e is f = log e. This logarithm occurs so often that it is called the natural logarithm and is abbreviated as ln. Whenever ou see ln it means log e. = 3 = e Definition 5.3 The function f = e is called the natural eponential function. Its inverse is called the natural logarithm and is denoted as ln, that is, ln = log e. Figure 5.3 shows the graph of both e and ln. Because the are inverses of one another, the graph of one is the reflection of the other s graph across the line =. Take note that the positive real numbers constitute the domain of ln. Its range if R.
7 Eponential and Logarithmic Functions = e = ln 7 6 5 4 3 3 3 4 5 6 7 = ln Figure 5.3. The natural eponential function e and its inverse ln Never forget the meaning of ln. It is log e, which we also call e. Its meaning is simpl ln = e = power for which e =. 5.6 For eample, ln = e = 0 because e 0 =. Also lne = e e = because putting in the bo gives e. In general lne = e e =, so ln ln e = ln e = e e =, ln e = ln e / = e e / = 3 e, = ln e 3/ = e e 3/ = 3, lne = e e =. This last equation lne = is simpl f f = for the case f = e. Of course we can t alwas work out ln mentall, as was done above. If confronted with, sa, ln9 we might think of it as e 9 but we re stuck because no obvious number goes in the bo to give 9. The best we can do is guess that because e.7 is close to 3 and 3 = 9, then ln9 must be close to. Fortunatel, a good calculator has a ln button. Using it, we find ln9.97.
The Natural Eponential and Logarithm Functions 73 Although the e notation ma be a helpful nemonic device, we will not use it etensivel, and ou will probabl drop it ourself once ou have become fluent in logarithms. Perhaps that has alread happened. Please be aware that it is common particularl in other tets to drop the parentheses and write ln simpl as ln. For clarit this tet will mostl retain the parentheses, we will certainl drop them occasionall. Because ln = log e, all standard logarithm properties appl to it. Natural Logarithm Laws lne = ln = 0 ln = ln + ln ln r = r ln e ln = lne = ln = ln ln ln = ln Eample 5.5 e Simplif ln5 + ln. 5 B natural log laws this is ln5 + lne ln5 = ln5 + lne ln5 = lne = =. The law ln r = r ln is etremel useful because it means taking ln of r converts the eponent r to a product. Consequentl ln can be used to solve an equation for a quantit that appears as an eponent. Eample 5.6 Solve the equation 5 +7 =. In other words we want to find the value of that makes this true. Since occurs as an eponent, we take ln of both sides and simplif with log laws. ln 5 +7 = ln + 7 ln5 = ln ln5 + 7ln5 = ln ln5 ln = 7ln5 ln5 ln = 7ln5 = 7 ln5 ln5 ln.9595585, where we have used a calculator in the final step. In doing this problem we could have used a logarithm to an base, not just ln = log e. Most calculators have a log button for the common
74 Eponential and Logarithmic Functions logarithm log 0. With log the steps above gives the same solution = 7log5 log5 log.9595585. The net two eamples will lead to a significant formula called the change of base formula. Eample 5.7 Suppose b is positive. Solve the equation b = for. The variable is an eponent, so we take ln of both sides and simplif. ln b = ln lnb = ln = ln lnb Therefore, in terms of and b, the quantit is the number ln lnb. Eample 5.7 would have been quicker if we had used log b instead of ln. Lets s do the same problem again with this alternative approach. Eample 5.8 Suppose b is positive. Solve the equation b = for. The variable is an eponent, so we take log b of both sides and simplif. log b b = log b = log b Therefore, in terms of and b, the quanitt is the number log b. Eamples 5.7 and 5.8 sa the solution of b = can be epressed either as ln lnb or log b. The first solution ma be preferable, as our calculator has no log b button. But what is significant is that these two methods arrive at the same solution, which is to sa log b = ln lnb. To summarize: Change of Base Formula log b = ln lnb The change of base formula sas that a logarithm log b to an base b can be epressed entirel in terms of the natural logarithm ln.
The Natural Eponential and Logarithm Functions 75 Eample 5.9 B the change of base formula, log 5 = ln5 ln.6094379 0.69347 =.3980. This seems about right because log 5 = 5 is the number for which = 5. Now, = 4 < 5 < 8 = 3, so should be between and 3. This eample shows in fact =.3980 to seven decimal places. Eercises for Section 5.5 Simplif the following epressions.. ln. ln7e ln49 3. e e e 4. ln6 ln4 ln9 5. e ln3 ln 6. e 5ln3 Use the techniques of this section to solve the following equations. 7. +3 = 3 8. 0 4 = 9. e 4 = e 0. e + =. e sin = e. sin = cos 3. 0 = 7 4. ln + 5 + 7 = 0 Use the change of base formula to epress the following logarithms in terms of ln. 5. log 5 6. log 3 5 7. log 4 5 8. log 5 5 9. log8 0. log9. log0. log99 3. log 3 8 4. log 3 9 5. log 3 0 6. log 3 9
76 Eponential and Logarithmic Functions 5.6 Wh e is the Best Base What makes e so special? Good answers will come later. We will see that the calculus of eponential functions is easier, simpler, and neater with base e. But wh would we epect this? The net two pages show that in a ver real sense e is the most efficient base. If ou don t want to know, then skip this section; it s not used elsewhere. The familiar base-0 number sstem uses ten smbols 0,,,3,...,9. An integer can be broken down into a sum of multiples of powers of 0. For eample, 7436 = 7 0 3 + 4 0 + 3 0 + 6 0 0. For an positive integer n there is a base-n number sstem. Some are noteworth. Internall, computers use the base-, or binar number sstem having onl two digits, 0 and. In base the number nineteen sa is written 00 because nineteen is 4 + 0 3 + 0 + + 0. The ancient Bablonians used a base-60 number sstem with 60 different cuneiform digits including a blank, which the used for what we call 0. In base n there are a total of n k numbers of length k or less. For eample, in base 0 four or fewer digits can epress 0 4 numbers, from 0 to 9999. With k digits we get 0 k numbers. So the Bablonians had more number smbols than we do, but their numbers were shorter. The got 60 k numbers from k digits. We get 0 k. Is there a best or optimal base? Think phones. A phone uses base 0. It has 0 buttons one per digit and a 0-digit readout can displa 0 0 = 0,000,000,000 phone numbers. If phones used base, a phone would have just two buttons 0 and. But to get 0 0 different phone numbers, each number would have to be 34 digits long. Because k = 34 is the smallest eponent for which k 0 0. 3 456 7890 3 4 5 6 7 8 9 0 000 0 0 Your Phone Ten buttons, and phone numbers are 0 digits long. Binar Phone Two buttons, but phone numbers are 34 digits long. Bablonian Phone Sit buttons, but their phone numbers were onl 6 digits long.
Wh e is the Best Base 77 In general, a base-n phone would have n buttons. To have 0 0 possible phone numbers, each phone number would have to be k-digits long, where k is the least integer with n k 0 0. Taking log 0 of both sides gives log 0 n k log 0 0 0, or k log 0 n 0, so k 0 log 0 n. Conclusion: If phones used base n and there were 0 0 possible phone numbers, each phone number would have to be k = 0 log 0 n digits long rounded up. For the base-60 Bablonians, k = 0 log 0 60 5.6, so their phone numbers would be onl 6 digits long. There is a trade-off. Lower bases n result in fewer smbols for digits and fewer buttons but the length k = 0 log 0 n required to epress a number of a given magnitude is greater. Ideall we would like both n and k to be small, but making one smaller forces the other larger. It seems reasonable, therefore, that the most efficient base is the n that makes the product n k = n 0 log 0 n as small as possible. To find it, let s graph the equation 0 = n log 0 n for various values of n and see which n results in the lowest value. = 0n log 0 n e 3 4 5 6 7 8 9 0 n number sstem base From the graph it appears that n = 3 is actuall the most efficient base for a number sstem. But the absolute lowest point on the graph is not at the integer n = 3, but rather somewhere around n =.7. Actuall it is possible to show using calculus which we will do that the lowest point occurs at eactl e =.78888. Thus e is the most efficient base! Of course this demonstration is sill because the base of a number sstem must be a positive integer. And because Bablonians didn t have phones. But it does suggest that there is something special about e as a base of an eponential function. You will see manifestations of this specialness throughout this book, and beond.