MA123, Supplement: Exponential and logarithmic functions (pp , Gootman)

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1 MA23, Supplement: Exponential an logarithmic functions pp , Gootman) Chapter Goals: Review properties of exponential an logarithmic functions. Learn how to ifferentiate exponential an logarithmic functions. Learn about exponential growth an ecay phenomena. Assignments: Assignment 0 Assignment Exponential notation: Quick review If a is any real number an n is a positive integer, then the n-th power of a is a n = a a... a }{{} n times The number a is calle the base whereas n is calle the exponent. The first an secon laws of exponents below allow us to efine a n for any integer n. Now, we want to efine, for instance, a /3 in a way that is consistent with the laws of exponents. We woul like: a /3 ) 3 = a /3)3 = a = a; thus a /3 = 3 a So, by the efinition of nth root, we have: Definition of rational exponents: For any rational exponent m/n in lowest terms, where m an n are integers an n > 0, we efine a m/n = a /n ) m = n a) m a m/n = a m ) /n = n a m or equivalently a /n = n a If n is even we require that a 0. Inthetablebelowthebasesaanbarerealnumbers 0ifneee)antheexponentsxanyarerationalnumbers. Laws of exponents:.) a 0 = 2.) a x = a x 3.) a x a y = a x+y 4.) ax a y = ax y 5.) a x ) y = a xy 6.) ab) x = a x b x ) a x 7.) = ax b b x Now, let a > 0 be a positive number with a. Thus far a x is efine for x a rational number. So, what oes, forinstance, 5 2 mean? Whenxis irrational, wesuccessively approximate x by rational numbers. For instance, as we successively approximate 5 2 with 5.4, 5.4, 5.44, 5.442, 5.442,... In practice, we simply use our calculator an fin out Graphs of exponential functions: The exponential function fx) = a x a > 0, a ) has omain R an range 0, ). The graph of fx) has one of these shapes: y y Exponential functions: Let a > 0 be a positive number with a. The exponential function with base a is efine by fx) = a x for all real numbers x. 0 x fx) = a x for a > 0 fx) = a x for 0 < a < x 5

2 The most important base is the number enote by the letter e. The number e is efine as e = lim + ) n n n Correct to five ecimal places note that e is an irrational number), e The natural exponential function: The natural exponential function is the exponential function fx) = e x with base e. It is often referre to as the exponential function. Since 2 < e < 3, the graph of y = e x lies between the graphs of y = 2 x an y = 3 x. y y = 3 x 0 y = 2 x y = e x x n + ) n n , , , ,000, Logarithmic functions: Every exponential function fx) = a x, with a > 0 an a, is a one-to-one function by the horizontal line test. Thus, it has an inverse function. The inverse function f x) is calle the logarithmic function with base a an is enote by log a x. Definition: Let a be a positive number with a. The logarithmic function with base a, enote by log a, is efine by y = log a x) a y = x. In other wors, log a x) is the exponent to which the base a must be raise to give x. Properties of logarithms:.) log a ) = 0 2.) log a a) = Since logarithms are exponents, the laws of exponents give rise to the laws of logarithms: 3.) log a a x ) = x 4.) a log a x) = x Let a be a positive number, with a. Let A, B an C be any real numbers with A > 0 an B > 0. Laws of logarithms:.) log a AB) = log a A)+log a B); ) A 2.) log a = log B a A) log a B); Change of base: For some purposes, we fin it useful to change from logarithms in one base to logarithms in another base. One can prove that: log b x = log ax) log a b) 3.) log a A C ) = C log a A). Remark: If a one-to-one function f has omain A an range B, then its inverse function f has omain B an range A. THUS, the function y = log a x) is efine for x > 0 an has range equal to R. More precisely: Graphs of logarithmic functions: Thegraphoff x) = log a x)isobtainebyreflecting the graph of fx) = a x in the line y = x. The picture below shows a typical case with a >.) y 0 y = 2 x y = x y = log 2 x) x The point,0) is on the graph of y = log a x) as log a ) = 0) an the y-axis is a vertical asymptote. 52

3 Common logarithms: The logarithm with base 0 is calle the common logarithm an is enote by omitting the base: logx) := log 0 x). Natural logarithms: Of all possible bases a for logarithms, it turns out that the most convenient choice for the purposes of Calculus is the number e. Definition: The logarithm with base e is calle the natural logarithm an is enote by ln: lnx) := log e x). We recall again that, by the efinition of inverse functions, we have y = lnx) e y = x. Properties of natural logarithms:.) ln) = 0 3.) lne x ) = x 2.) lne) = 4.) e lnx) = x Derivatives e h Fact: By filling the table below we can convince ourselves that lim =. h 0 h h e h h Now, let fx) = e x. Using the efinition of the erivative an the rules for exponential functions, we have that fx+h) fx) e x+h e x x ex ) = lim = lim h 0 h h 0 h = lim h 0 e x e h e x h = e x lim h 0 e h h ) = e x Theorem: x ex ) = e x or e x ) = e x. Moreover, it follows by applying the chain rule that x egx) ) = e gx) x gx)) or egx) ) = e gx) g x). We can use the erivative of e x an the relationship between the exponential an the natural logarithmic functions to fin the erivative of the function lnx). Namely, take the erivative with respect to x of both sies of e lnx) = x. We obtain x elnx ) = x x) or elnx x lnx) = or x lnx) = x. Theorem: x lnx)) = x or lnx)) = x. Moreover, it follows by applying the chain rule that x lngx))) = gx) x gx)) or ln gx))) = g x) gx). 53

4 What about more general erivatives? Observe that we have the ientities a x = e lnax) = e xlna) log a x) = lnx) lna). Thus using the previous results we obtain the following formulas for the erivatives of general exponential an logarithmic functions Note: x ax ) = a x lna) an x log ax)) = xlna). Let us consier the function fx) = 3 x. In Example 6 of Chapter 4, we saw that an approximation for f ) was given by the value Using the above formula we have that f x) = 3 x ln3), so that the exact value for f ) is 3ln3) = ln27). Example : Fin the erivative with respect to x of fx) = e 4x. Evaluate f x) at x = /4. Compute f x), f x) an f 0) x). Can you guess what the nth erivative f n) x) of fx) looks like? Example 2: Fin the erivative with respect to x of gx) = x 2 e x. Evaluate g x) at x =. Example 3: Suppose ft) = e 3t 4. Fin f t. Example 4: Fin the erivative with respect to x of y = lne x ). 54

5 Example 5: Fin the erivative with respect to x of fx) = xlnx). Example 6: Fin the erivative with respect to x of y = ln5x+). Example 7: Fin ) ln3x 4 7x 2 +5). x Example 8: Fin the erivative with respect to x of fx) = lnlnlnx))). Example 9: Fin the erivative with respect to x of hx) = e x2 +3lnx). 55

6 Exponential growth an ecay Let Qt) enote the amount of a quantity as a function of time. We say that Qt) grows exponentially as a function of time if Qt) = Q 0 e rt, where Q 0 an r are positive constants that epen on the specific problem an t enotes time. When t = 0, we see that Q0) = Q 0 e r 0 = Q 0 = Q 0. Thus Q 0 enotes the amount of the quantity at t = 0. In other wors, Q 0 is the initial amount of the quantity at time t = 0. Note that Qt) > 0 because Q 0 > 0 an e rt > 0. Taking the erivative an using the chain rule, we see that Q t) = Q 0 r e rt = rq 0 e rt ) = rqt). Since Q t) = rqt), it follows that if a quantity grows exponentially, then its rate of growth is proportional to the quantity present, an the proportionality constant is given by r. Since r > 0 an Qt) > 0, we have Q t) > 0, as expecte because Qt) is increasing. Some quantities ecrease exponentially. In this case we have Qt) = Q 0 e rt, where Q 0 an r are positive constants. Note that we have Q0) = Q 0 an Q t) = Q 0 r) e rt = rq 0 e rt ) = rqt). Thus Q t) = rqt). We see that Q t) < 0 because r < 0 an Qt) > 0. Thus the rate of increase of Qt) is proportional to the quantity present, an the proportionalityconstant is given by r. Suppose that a function gx) satisfies the property that the slope of the tangent line to the graph of y = gx) at any point P is proportional to the y-coorinate of P, i.e., g x P ) = r gx P ). Then it can be shown that there are constants C an r such that gx) = Ce rx. In fact, r is the constant of proportionality because g x) = rce rx = rgx). Example 0: The graph of a function gx) passes through the point 0,5). Suppose that the slope of the tangent line to the graph of y = gx) at any point P is 7 times the y-coorinate of P. Fin g2). 56

7 Applications Many processes that occur in nature, such as calculation of bank interest, population growth, raioactive ecay, heat iffusion, an numerous others, can be moele using exponential functions. Logarithmic functions are use in moels for the louness of souns, the intensity of earthquakes, an many other phenomena. where Compoun interest is calculate by the formula: Pt) = P 0 + r ) nt n Pt) = principal after t years P 0 = initial principal r = interest rate per year n = number of times interest is compoune per year t = number of years Continuously compoune interest is calculate by the formula: Pt) = P 0 e rt where Pt) = principal after t years P 0 = initial principal r = interest rate per year t = number of years Proof: The interest pai increases as the number n of compouning perios increases. If m = n r, then: P 0 + r ) nt [ = P 0 + r ) n/r ] rt [ = P 0 + ) n/r ] rt [ = P 0 + m ] rt. n n n/r m) As n becomes large, m also becomes large. Since lim + m = e we obtain the formula for continuously m m) compoune interest. Example : If $0,000 is investe at an interest rate of 6%, fin the value of the investment at the en of 8 years if the interest is compoune continuously. Example 2: How many years will it take an investment to quaruple in value if the interest is compoune continuously at a rate of 7%? 57

8 Example 3: An amount of P 0 ollars is investe at 5% interest compoune continuously. Fin P 0 if at the en of 0 years the value of the investment is $20,000. Exponential moels of population growth: The formula for population growth of several species is the same as that for continuously compoune interest. In fact in both cases the rate of growth of a population or an investment) per time perio is proportional to the size of the population or the amount of an investment). Remark: Biologists sometimes express the growth rate r in terms of the oubling-time t 0, the time require for the population to ouble in size: r = ln2). t 0 Exponential growth moel If P 0 is the initial size of a population that experiences exponential growth, then the population Pt) at time t increases accoring to the moel Pt) = P 0 e rt where r is the relative rate of growth of the population expresse as a proportion of the population). Note: If t 0 enotes the oubling-time of a population, we can rewrite the expression for Pt) as follows Pt) = P 0 e rt = P 0 e ln2)/t 0) t = P 0 e ln2) ) t/t0 = P 0 2 t/t 0. Example 4: A bacteria culture starts with 2, 000 bacteria an the population triples after 5 hours. If we express the number of bacteria after t hours as yt) = ae bt fin a an b. Example 5: A bacteria culture starts with 5, 000 bacteria an the population quaruples after 3 hours. Fin an expression for the number Pt) of bacteria after t hours. 58

9 Example 6: If the bacteria in a culture oubles in 3 hours, how many hours will it take before 7 times the original number is present? Example 7: If the worl population in 200 was 6 billion an it were to grow exponentially with a growth constant r = ln2), fin the population in billions) in the year Raioactive ecay: Raioactive substances ecay by spontaneously emitting raiation. In this situation, the rate of ecay is proportional to the mass of the substance. This is analogous to population growth, except that the quantity of raioactive material ecreases. Remark: Physicists sometimes express the rate of ecay in terms of the half-life, the time require for half the mass to ecay. Raioactive ecay moel: If Q 0 is the initial quantity of a raioactive substance with half-life t 0, then the quantity Qt) remaining at time t is moele by the function where r = ln2) t 0. Qt) = Q 0 e rt Note: If t 0 enotes the half-life of a raioactive substance, we can rewrite the expression for Qt) as follows Qt) = Q 0 e rt = Q 0 e ln2)/t 0) t = Q 0 e ln2) ) t/t0 = Q 0 2 t/t 0 = Q 0 2 ) t/t 0 = Q 0 2) t/t0. Example 8: The half-life of Cesium-37 is 30 years. Suppose we have a 00 gram sample. How much of the sample will remain after 50 years? 59

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