**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