MA 2232 Lecture 08 - Review of Log an Exponential Functions an Exponential Growth Friay, February 2, 2018. Objectives: Review log an exponential functions, their erivative an integration formulas. Exponential Growth Exponential Functions Repeate multiplication is the basic iea behin the use of exponents. For example, (1) 2 4 = 2 2 2 2 = 16. It turns out that it kin of makes sense to exten the efinition of exponents in certain ways, an things work out really well. By efinition, we take the following expressions to mean (2) (3) 2 0 = 1 2 3 = 1 2 3 (4) 2 1/2 = 2, an these allow us to make sense of any exponent that is a fraction. Amazingly, all of the important properties that exponents have still work uner the extene efinitions. In fact, if you graph the values of the function f(x) = 2 x for the fractional values of x, they all lie on a nice, smooth curve. It s har to make sense of an irrational exponent, like 2 π, algebraically, but since the rational (i.e. fractional) exponents graph nicely, we can exten to irrational exponents in terms of limits. What I m saying is that our exponential functions are mae-up functions that exten our basic unerstaning of exponents in the most natural way possible. Log Functions Once we have a function like f(x) = 2 x efine, it is convenient to have an inverse function available. An inverse function, in some sense, reverses the x s an y s. The name we use for the inverse of 2 x is log 2 (x). For example, if (5) y = f(3) = 2 3 = 8, then g(x) = log 2 (x) is efine so that (6) y = g(8) = log 2 (8) = 3. 1
MA 2232 Lecture 08 - Review of Log an Exponential Functions an Exponential Growth 2 In general, (7) (8) 2 log 2(x) = x, an log 2 (2 x ) = x. The functions 2 x an log 2 (x) uno each other. The graph of log 2 (x) has exactly the same shape as 2 x, except with the x s an y s reverse, which flips the graph over on its sie. For every positive real number a, we have an exponential function a x an log a (x), an they are all relate to each other in the same way (9) (10) The graphs all have the same basic shape. a log a (x) = x, an log a (a x ) = x. Solving Equations with Exponential an Log Functions One of the reasons for wanting pairs of inverse functions is algebraic. When we solve equations, we manipulate them by unoing things we on t want. To get ri of a multiplie constant, we ivie both sies of the equation by that number. Multiplication an ivision uno each other. Consier the equation (11) 7 2x 1 = 12. To solve for x, we want to get x by itself on one sie of the equation, an we o that by getting ri of the everything aroun the x. As it stans, x is insie of an exponential function f(x) = 7 x. The inverse of this exponential function is g(x) = log 7 (x), so we shoul apply this log function to both sies of the equation (12) log 7 ( 7 2x 1 ) = log 7 (12). The log function unoes the exponential function, an we have (13) 2x 1 = log 7 (12). To uno the minus 1, we a 1, an after that, to uno the times 2 we ivie by 2. This results in (14) x = log 7(12) + 1. 2 Using your calculator to compute a log. If we want a ecimal value for x, we ll nee to use our calculator. Most of our calculators only have a button for ln(x) = log e (x) an log 10 (x), but the following formula allows us to fin logs for any base (15) log a (x) = log b(x) log b (a).
MA 2232 Lecture 08 - Review of Log an Exponential Functions an Exponential Growth 3 In this last example, we have a = 7. Our calculators can o b = e or b = 10. Using natural log, we have (16) log 7 (12) = ln(12) ln(7) = 1.276989408. The solution to the equation above woul be (17) x = log 7(12) + 1 2 = 1.138494704. Base e We ve been using this alreay, but remember that we have a special constant e, (18) e = 2.718281828459045235..., an we ll mostly o our exponential an log functions with e as the base. So THE exponential function will be e x, an the corresponing log function is the natural log function is the log base e, (19) ln(x) = log e (x). These, as you may recall, ha the nicest erivatives (20) an x [ ex ] = e x, (21) x [ln(x)] = 1 x. The erivatives of the other exponential an log functions So what are the erivatives of a x an log a (x)? Since we alreay know the erivatives of e x an ln(x), we shoul bring these in. Note that we can always reformulate our exponential an log problems in terms of the base e. Using (22) a = e ln(a). we can rewrite a x as follows, for any a. (23) a x = For completeness, we have the erivatives (24) (25) an base e is just a special case. ( e ln(a) ) x = e x ln(a), x [ ax ] = a x ln(a) x [ log a (x)] = 1 x ln(a) Note again that given any exponential function f(x) = a x, since we can write (26) a = e ln(a), we express f in terms of the exponential function e x. In particular, (27) f(x) = a x = e xln(a). For example, given the function f(x) = 3 x, we know that ln(3) = 1.09861228..., an so in applications, we will generally use something like (28) f(x) e 1.0986x instea. In other wors, we really only nee one exponential function, an we might as well use e x (a.k.a. exp(x)). Furthermore, if we re only going to use e x, then we ll also only nee one log function, log e (x) = ln(x).
MA 2232 Lecture 08 - Review of Log an Exponential Functions an Exponential Growth 4 Integral formulas The corresponing integral formulas are (29) e x x = e x + C, an (30) 1 x = ln(x) + C. x Absolute Value Signs? If you look at a table of integrals, you might see some absolute values in the natural log erivative an integral formulas. These just make the formulas more general, an we won t worry about them a lot. Here s an explanation, anyway. We have (31) x [ln(x)] = 1 x, but we shoul keep in min that since e x > 0 for all x, the natural log function is only efine for positive x s. We can see this in the graph of the natural log. If we were to take the absolute value of x before putting it in the natural log, that is, let f(x) = ln x, then this function woul be efine for all real numbers, except x = 0. The graph of this function looks like It s not completely clear in the picture, but the graph goes to negative infinity on either sie of the y-axis. Now. As you look at the graph of f(x) = ln( x ), the left branch is a mirror image of the right branch. The slopes are the same, except they re negative. Since 1 x is also negative for negative x s, we still have that 1 x is equal to the slopes on f. In other wors, (32) x [ ln( x )] = 1 x is vali for all non-zero x s.
MA 2232 Lecture 08 - Review of Log an Exponential Functions an Exponential Growth 5 Exponential growth moels We can think of calculus as starting with limits an erivatives. The erivative of a function tells us the slopes of the tangent lines, which can be interprete as a rate of change, how fast y is changing relative to x. A slope of 3 means that y increases 3 for every increase of 1 in x. Next we look at integrals an antierivatives. These are a bit harer, but actually provie us with more information. Integral applications are much more common than ifferential ones. Even more useful sithe next step. The ifferential equation. A ifferential equation is an eequation involving an unknown function an its erivatives. Solving the equation, i.e. fining the function, is a generalization of fining an antierivative. Here is a simple example. The exponential growth moel (a moel is a function escribing some phenomenon) is base on the assumption that the growth rate P (t) is proportional to the population size P(t). For example, if a population grows by 3% this year, then it ll probably grow by 3% every year. It s not obvious, however, what the actual growth rate might be, but it s important to assume that there is some constant growth rate r. This leas us to the equation (33) P (t) = r P(t). This is an example of a ifferential equation, the unknown population function P(t) is in the equation along with its erivative, the population growth rate P (t). With some thought, you might come up with a solution. (34) P(t) = e rt, an this clearly works. More generally, we see that (35) P(t) = Ce rt is a solution for all C (this is kin of like the +C in our inefinite integrals). Before we start with examples, note that (36) P(0) = Ce r 0 = C, an so the constant C is the population size at time t = 0. We ll call this the initial population size. Example. Suppose we have a cell culture that initially consists of 520 cells, an after three hours, has 610. Note that this kin of information woul be easily obtainable. If the cell culture is free to grow, an therefore, nothing will cause the growth rate to slow own (an not be constant), we can assume exponential growth. Since the initial population size is C = 520, we have the growth function (37) P(t) = 520e rt. The one thing we like to know is the growth rate r. We can solve for r using the fact that at time t = 3, the population size is 610. In other wors, (38) P(3) = 610. We can also express P(3) with our exponential growth function as (39) P(3) = 520e r 3. This gives us the equation (40) 610 = 520e 3r.
MA 2232 Lecture 08 - Review of Log an Exponential Functions an Exponential Growth 6 We woul like to solve for r. The key to oing this is remembering that ln(x) is the inverse function for e x, an ln(e x ) = x. We ll solve the equation like this (41) (42) (43) (44) (45) Our function, therefore, is 610 = 520e 3r 610 520 = e3r ( ) 610 ln = ln ( e ) 3r = 3r 520 ( ) 1 610 3 ln = r. 520 0.053210 = r (46) P(t) = 520e 0.053210r. From here, we can preict the population size at any time we want. In 10 hours, for example, we woul have (47) P(10) = 520e 0.053210 10 = 885.30 885. Homework 08 1. Suppose you introuce 150 rats onto an islan, an there is plenty of foo an room for the population of rats to expan. You can assume exponential growth in this situation. After 50 ays, you somehow fin that there are 330 rats. a. Fin C an r in the growth function P(t) = Ce rt. b. Preict the population size at t = 100 ays. c. One of the things were assuming in an exponential growth moel is the fact that the growth rate will stay the same. Can you think of anything normal that will make the growth rate change an shoul be accounte for in the moel? 2. Suppose we have a bacterial culture that starts with 57 cells, an after 2 hours has 83 cells. Fin C, r, an the number of cells after one ay (24 hours). Your t shoul be measure in hours.
MA 2232 Lecture 08 - Review of Log an Exponential Functions an Exponential Growth 7 Nothing in the exponential growth moel ifferential equation requires r to be positive. An the population coul be any number of things (like money, amount of oxygen in a room, etc.). 3. For example, if you ha some object that containe 1,000 carbon-14 atoms, because of raio-active ecay, after about 5,730 years, you ll only have about 500 carbon-14 atoms left. Fin C, r, an the amount left after 700 years. Roun your r to eight ecimal places this time, an note that it will be negative. Note: Living things have essentially the same concentration of carbon-14 as the atmosphere, but after they ie, the carbon-14 is not replace. Using the function you just foun, you can figure out about when something was alive. Answers: 1) C = 150, r = 0.015769, P(100) = 726 (roune to whole numbers) 2) C = 57, r = 0.1879, P(24) = 5180 (roune to whole numbers) 3) C = 1,000, r = 0.00012097, P(700) = 919 (roune to whole numbers)