Chapter 8

March 22 2017

Chapter 8 Exponential and Logarithm Functions 8.1 Introduction - our main purpose in this chapter is learn how to work successfully with the indefinite integral S dx / x - as we shall see, this purpose compels us to study the special exponential and logarithm functions - y = ex and y = log e x - the letter e used in these functions denotes the most important special number in mathematics after pi - in decimal form it is an infinite nonrepeating decimal that is known to hundreds of thousands of decimal places (2.7182 ...) - the ultimate reason for our interest in these matters is that the integral and the functions arise in a great variety of problems - in order to reach a clear understanding of why the number e and the functions matter so much - is is desirable to broaden the context a bit and consider the more general exponential and logarithm functions - y = ax and y = log a x - where a is a positive constant != 1 - this is where we begin - and by adopting this approach we hope to make it perfectly clear that we choose a equal to e for reasons of convenience and simplicity 8.2 Review of Exponents and Logarithms - if r = p/q is a fraction in lowest terms with q > 0, then by definition - a r = a p/q = (sqrt q a) p - where sqrt q a is the unique positive number whose qth power is a - if the exponent x is an irrational number - then difficulties appear that students might not notice if we didn't mention them - for instance, what is meant by the expression 2 sqrt 2? - clearly it doesn't make sense to multiple 2 by itself sqrt 2 times - also, since sqrt 2 can't be written as a fraction - the definition a r = a p/q is useless - is 2 sqrt 2 really a definite number with a specific value? - the answer is yes, but this is not at all obvious - a natural way to proceed is to use the fact that any irrational number can be approximated as closely as we please by rational numbers - we can therefore define a x by a x = lim r>x a r - where r approaches x through rational values - it can be proved that the laws of exponents a x1 a x2 = a x1+x2 continue to hold for arbitrary real numbers of x - the next natural step in this development is to examine the properties of the general exponential function y = ax - here again, we simply state the important facts without making any attempt to discuss the logical details of how these facts are established - as above, we assume that a is a positive constant - and also that a != 1 - the case a = 1 is of no interest because 1 x = 1 for all x - let us suppose first that a > 1 - then y = ax is a continuous function of x, it is increasing, its values are all positive - and lim x > -inf a x = 0 and lim x > inf a x = inf - if a < 1 - then y = ax is a decreasing function - when this much information about exponents is known or assumed - it is very easy to define logarithms and obtain some of their properties - on the most primitive level, a logarithm is an exponent - thus, the fact that 100 = 10 2 says that 2 is the logarithm of 100 to the base 10 - written 2 = log 10 100 - and 4 = 64 1/3 says that 1/3 is the logarithm of 4 to the base of 64 - 1/3 = log 64 4 - more generally, the properties of exponents discussed above show clearly that if a is a positive constant != 1 - then to each positive x there corresponds a unique y such that x = ay - this y is written in the form y = log a x - and is called the logarithm of x to the base a - accordingly, y = log a x has the same meaning as x = a y - in the sense that each equation expresses the same relation between x and y - with the first written in a form solved for y - and the second in a form solved for x - we can state this somewhat differently by saying that - the symbol "log a" is created for the specific purpose of enabling us to solve x = ay for y in terms of x - the basic properties of logarithms are direct translations of corresponding properties of exponents - so if x1 = a y1 and x2 = a y2 - then x1 x2 = a y1 a y2 = a y1+y2 - but y1 = log a x1 and y2 = log a x2, so we have - log a x1x2 = log a x1 + log a x2 - similarly, log a x1/x2 = log a x1 - log a x2 - and log a xb = b log a x - also a log a x = x and log a a x = x - we note also the particular facts log a 1 = 0 and log a a = 1 - are equivalent to 1 = a 0 and a = a 1 - in studying the logarithm function y = log a x - we consciously think of x and y as variables instead of mere numbers - our starting point is the fact that y = log a x is equivalent to x = a y - it is clear from this that x must be positive in order for y to exist - so y = log a x is defined only for x > 0 - the graph of y = log a x is easy to obtain from the graph of x = ay by interchanging the axes - in this case y = log a x is evidently an increasing continuous function of x - the features of this function that correspond to the properties of the limits of ax are - lim x>0+ log a x = -inf and lim x>inf log a x = inf - the most convenient logarithm for actual numerical calculations is the logarithm to the base 10 - the so-called common logarithm - the importance of the logarithm as a function is evident as it remains indispensable in the theoretical pats of mathematics and its applications - these theoretical uses are what concern us in this chapter 8.3 The Number e and the Function y = e x - the number e is often defined by the limit - e = lim n>inf (1 + 1/n) n - this definition has the advantage of brevity but the serious disadvantage of shedding no light whatever on the significance of this crucial number - we prefer to define e differently, in a manner that reveals as clearly as possible why this number is so important - we then obtain the above definition later, as merely one among many explicit formulas for e that can be used in a variety of ways - our aim in this section is to study a function y = f(x) that is unchanged by differentiation - d/dx f(x) = f(x) - it is far from obvious that any such function exists - as we shall see, the desired function turns out to be one of the exponential functions y = a x for a > 1 - the central meaning of the number e can be stated as follows - it is the specific value of the base a that causes the function f(x) = a x to be unchanged by differentiation - in this way we understand what purpose e serves - however, we must still give a satisfactory definition and show as simply as possible that this definition accomplishes the stated purpose - let us calculate the derivative of f(x) and see what happens - as usual when differentiating a new type of function - we must go back to the definition of the derivative - d/dx f(x) = lim dx>0 f(x + dx) - f(x) / dx - it will be convenient here to denote the increment by the single letter h instead of the familiar dx - d/dx a x = a x ( lim h>0 ah - 1 / h ) - as a graph of y = a x shows - the quantity a h - 1 / h is the slope of the tangent line to the curve y = a x at point (0, 1) - if this slope equals 1, then the right side reduces to a x and this particular function has the property d/dx f(x) = f(x) - this brings us to the definition, e is the specific value of the base a that produces this result - that is, e is the number for which lim h>0 eh - 1 / h = 1 - we can obtain considerable insight into the nature of the number e by sketching y = a x for the cases a = 1.5, 2, 3 and 10 - these curves tell us that as the base a increases continuously from numbers close to 1 to larger numbers - the slope of the tangent to y = a x at the point (0, 1) increases continuously from values close to 0 to larger values - therefore this slope is exactly equal to 1 for some intermediate value of a - this intermediate value is e - it is geometrically clear from these remarks that e exists - y = e x is the single member of the family of exponential functions y = a x (a > 1) whose tangent line at point (0, 1) has slope 1 - the function y = e x is often called the exponential function, to distinguish it from its comparatively unimportant relatives - we can investigate the number e more closely by noting that eh - 1 / h is approximately equal to 1 - and this approximation gets better and better as h approaches 0 - by simple manipulations, we obtain - e = lim h>0 (1 + h) 1/h - in words, this says that e is the limit of 1 plus a small number - raised to the power of the reciprocal of the small number - as that small number approaches 0 - if we write h = 1 / n where n is understood to be a positive integer that > inf as h > 0, then - e = lim n>inf (1 + 1/n) n - this formula enables us to compute rough approximations to e fairly easily - the number e, like the number pi, is woven inseparably into the fabric of both nature and mathematics - many remarkable properties of e have been documented over the centuries - for example, e is irrational - indeed, it is not even the root of any polynomial equation with rational coefficients - however, we must not forget our original purpose in this section - which was to study a function that is unchanged by differentiation - we have now made a good start on this task - in the sense that we have explored the meaning of the following statement and established its validity - d/dx ex = ex - the equivalent statement is that y = ex satisfies the differential equation - dy/dx = y - every function y = cex also satisfies this equation - further, we assert that these are the only functions that are unchanged by differentiation - by the chain rule - d/dx eu = deu/du du/dx = eu du/dx - where u = u(x) is understood to be any differentiable function of x - if we write this in differential form - d(eu) = eu du - then by reading this backwards we obtain the integration formula - S eu du = eu + c - continuously compounded interest produces steady continuous growth of a type called exponential growth - A = P ext - in section 8.5 and 8.6 we discuss many additional examples of exponential growth as it occurs in the natural sciences 8.4 The Natural Logarithm Function y = ln x - logarithms to the base 10, common logarithms, are often taught in high school - starting with the following familiar definition - for any positive number x - log 10 x is that number y such that x = 10 y - in just the same way, for any positive number x, log e x is defined to be that number y such that x = e y - the number log e x is called the natural logarithm of x - for reasons that will become clear - in deference to standard practice at this level we denote this number by the simple notation ln x - y = ln x has the same meaning as x = e y - in the sense that we are dealing here with a single equation - first written in a form solved for y and then written in a form solved for x - the graph of y = ln x is obtained by simply turning over the graph of x = e y so as to interchange the positions of the axes - just as in section 8.2 - the natural logarithm function y = ln x is defined only for positive values of x and has the following familiar properties - ln x1 x2 = ln x1 + ln x2 - ln x1 / x2 = ln x1 - ln x2 - ln x b = b ln x - e ln x = x, ln e x = x - lim x>0+ ln x = -inf, lim x>inf ln x = inf - ln 1 = 0, ln e = 1 - we can compute the derivative of dy/dx of the function y = ln x very easily - by differentiating x = ey implicitly with respect to x - x = ey, 1 = ey dy/dx so dy/dx = 1/ey = 1/x - this yields the formula d/dx ln x = 1/x - and we immediately have the chain rule extension - d/dx ln u = 1/u du/dx - where u is understood to be any function of x - the differential version of the equation is d(ln x) = du/u - which leads at once to the main formula of this chapter - S du/u = ln u + c - it is understood that u is positive - because only in this case does ln u have a meaning - most of the applications require a quick transition from logs to exponentials - in situations where u is negative - we easily make minor adjustments by juggling the signs - students will recall that the fundamental integration formula - S un du = u n+1 / n+1 + c - failed to cover one exceptional case, namely, n = -1 - the formula above now fills this gap - since it tells us that - S u-1 du = S du/u = ln u + c - in section 5.4 we discussed the method of separation of variables for solving differential equations - the equation dy/dx = ky - is one of the simplest and most important to which this method can be applied - we give the details of this procedure here because the same idea will be used over and over again in the next two sections - the sooner students become thoroughly familiar with them, the better - dy/y = k dx, S dy/y = S kdx, ln y = kx + c1 - y = e kx + c1 = e c1 e kx - y = c e kx - where c is simply a more convenient notation for the constant e c1 - from our point of view, the exponential and logarithm functions fund their main reason for being in the fact that they enable us to solve the differential equation dy/dx = ky in this smooth and straight forward manner - it is also clear from the calculations that these functions go together like the two sides of a coin - the next two sections are filled with many far-reaching applications of - dy/dx = k dx, y = c e kx - to various fields of science - we hope students will agree that these applications fully justify the attention we have given to this differential equation and to the functions that are necessary for solving it 8.5 Applications. Population Growth and Radioactive Decay - as we emphasized in section 8.1 - our main purpose in this chapter is to develop the mathematical machines that is necessary for treating a variety of related applications - this machinery is now in place, and the time has come to see what it can do - exponential growth, population growth - dN / dt = kN - N = N0 e kt - exponential decay, radioactive decay - dx / dt = - kx - x = x0 e -kt - the positive constant k is called the rate constant 8.6 More Applications - when the rate of change is related to the current quantity - solving this relationship by separation of variables and dQ / Q - the solution of which invariably leads to the natural logarithm ln and the exponential function e