What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)
CHAPTER VIII. THE CALCULUS
§6. THE EXPONENTIAL FUNCTION AND THE LOGARITHM
The basic concepts of the calculus furnish a much more adequate theory of the logarithm and the exponential function than does the “elementary” procedure that underlies the usual instruction in school. There one usually begins with the integral powers an of a positive number a, and then defines , thus obtaining the value of ar for every rational r = n/m. The value of ax for any irrational x is next defined so as to make ax a continuous function of x, a delicate point which is omitted in elementary instruction. Finally, the logarithm of y to the base a,
x = loga y,
is defined as the inverse function of y = ax.
In the following theory of these functions on the basis of the calculus the order in which they are considered is reversed. We begin with the logarithm and then obtain the exponential function.
1. Definition and Properties of the Logarithm. Euler’s Number e
We define the logarithm, or more specifically the “natural logarithm,” F(x) = log x (its relation to the ordinary logarithm to the base 10 will be shown in Article 2), as the area under the curve y = 1/u from u = 1 to u = x, or, what amounts to the same thing, as the integral
(1)
(see Fig. 5, p. 29). The variable x may be any positive number. Zero is excluded because the integrand 1/u becomes infinite as u tends to 0.
It is quite natural to study the function F(x). For we know that the primitive function of any power xn is a function xn+1/(n + 1) of the same type, except for n = –1. In the latter case the denominator n + 1 would vanish and formula (4), p. 440 would be meaningless. Thus we might expect that the integration of 1/x or 1/u would lead to some new—and interesting—type of function.
Although we consider (1) the definition of the function log x, we do not “know” the function until we have derived its properties and have found means for its numerical computation. It is quite typical of the modern approach that we start with general concepts such as area and integral, establish definitions such as (1) on this basis, then deduce properties of the objects defined and, only at the very end, arrive at explicit expressions for numerical calculation.
The first important property of log x is an immediate consequence of the fundamental theorem of §5. This theorem yields the equation
(2) F′(x) = 1/x.
From (2) it follows that the derivative is always positive, which confirms the obvious fact that function log x is a monotone increasing function as we travel in the direction of increasing values of x.
The principal property of the logarithm is expressed by the formula
(3) log a + log b = log (ab).
The importance of this formula in the practical application of logarithms to numerical computations is well known. Intuitively, formula (3) could be obtained by looking at the areas defining the three quantities log a, log b, and log (ab). But we prefer to derive it by a reasoning typical of the calculus: Together with the function F(x) = log x we consider the second function
k(x) = log (ax) = log w = F(w),
setting w = f(x) = ax, where a is any positive constant. We can easily differentiate k(x) by rule (e) of §3: k′(x) = F′(w)f′(x). By (2), and since f′(x) = a, this becomes
k′(x) = a/w = a/ax = 1/x.
Therefore k(x) has the same derivative as F(x); hence, according to page 438, we have
log (ax) = k(x) = F(x) + c,
where c is a constant not depending on the particular value of x. The constant c is determined by the simple procedure of substituting for x the specific number 1. We know from the definition (1) that
F(1) = log 1 = 0,
because the defining integral has for x = 1 equal upper and lower limits. Hence we obtain
k(1) = log (a·1) = log a = log 1 + c = c,
which gives c = log a, and therefore for every x the formula
(3a) log (ax) = log a + log x.
Setting x = b we obtain the desired formula (3).
In particular (for a = x), we now find in succession
(4)
log (x2) = 2 log x
log (x3) = 3 log x.
...........................
log (xn) = n log x.
Equation (4) shows that for increasing values of x the values of log x tend to infinity. For the logarithm is a monotone increasing function and we have, for example
log (2n) = n log 2,
which tends to infinity with n. Furthermore we have
so that
(5)
Finally,
(6) log xr = r log x
for any rational number . For, setting xr = u, we have
so that
Since log x is a continuous monotone function of x, having the value 0 for x = 1 and tending to infinity as x increases, there must be some number greater than 1 such that for this value we have log x = 1.
Fig. 277.
Fig. 278.
Following Euler, we call this number e. (The equivalence with the definition of p. 298 will be shown later.) Thus e is defined by the equation
(7) log e = 1.
We have introduced the number e by an intrinsic property which assures its existence. Presently we shall carry our analysis further, obtaining as a consequence explicit formulas giving arbitrarily exact approximations to the numerical value of e.
2. The Exponential Function
Summarizing our previous results, we see that the function F(x) = log x has the value zero for x = 1, increases monotonically to infinity but with decreasing slope 1/x, and for positive values of x less than 1 is given by the negative of log 1/x, so that log x becomes negatively infinite as x → 0.
Because of the monotone character of y = log x we may consider the inverse function
x = E(y),
whose graph (Fig. 278) is obtained in the usual way from that of y = log x (Fig. 277), and which is defined for all values of y between – ∞ and + ∞. As y tends to – ∞ the value E(y) tends to zero, and as y tends to + ∞ E(y) tends to + ∞.
The E-function has the following fundamental property:
(8) E(a)·E(b) = E(a + b)
for any pair of values a and b. This law is merely another form of the law (3) for the logarithm. For if we set
E(b) = x, E(a) = z (i.e. b = log x, a = log z),
we have
log xz = log x + log z = b + a,
and therefore
E(b + a) = xz = E(a)·E(b),
which was to be proved.
Since by definition log e = 1, we have
E(1) = e,
and it follows from (8) that e2 = E(1)E(1) = E(2), etc. In general,
E(n) = en
for any integer n. Likewise , so that ; hence, setting p/q = r, we have
E(r) = er
for any rational r. Therefore it is appropriate to define the operation of raising the number e to an irrational power by setting
ey = E(y)
for any real number y, since the E-function is continuous for all values of y, and identical with the value of ey for rational y. We can now express the fundamental law (8) of the E-function, or exponential function, as it is called, by the equation
(9) eaeb = ea+b,
which is thereby established for arbitrary rational or irrational a and b.
In all these discussions we have been referring the logarithm and exponential function to the number e as a “base,” the “natural base” for the logarithm. The transition from the base e to any other positive number is easily made. We begin by considering the (natural) logarithm
α = log a,
so that
a = eα = elog a.
Now we define ax by the compound expression
(10) z = ax = eax = ex log a.
For example,
10x = ex log 10
We call the inverse function of ax the logarithm to the base a, and we see immediately that the natural logarithm of z is x times α; in other words, the logarithm of a number z to the base a is obtained by dividing the natural logarithm of z by the fixed natural logarithm of a. For a = 10 this is (to four significant figures)
log 10 = 2.303.
3. Formulas for Differentiation of e, ax, xs
Since we have defined the exponential function E(y) as the inverse of y = log x, it follows from the rule concerning differentiation of inverse functions (§3) that
i.e.
(11) E′(y) = E(y).
The natural exponential function is identical with its derivative.
This is really the source of all the properties of the exponential function and the basic reason for its importance in applications, as will become apparent in subsequent sections. Using the notation introduced in Section 2 we may write (11) as follows:
(11a)
More generally, differentiating the compound function
f(x) = eαx,
we obtain by the rule of §3
f′(x) = αeαx = αf(x).
Hence, for α = log a, we find that the function
f(x) = ax
has the derivative
f′(x) = ax log a.
We may now define the function
f(x) = xs
for any real exponent s and positive variable x by setting
xs = es log x.
Again applying the rule for differentiation of the compound functions, f(x) = esz, z = log x, we find and therefore
f′(x) = sxs–1,
in accordance with our previous result for rational s.
4. Explicit Expressions for e, ex, and log x as Limits
To find explicit formulas for these functions we shall exploit the differentiation formulas for the exponential function and the logarithm. Since the derivative of the function log x is 1/x, by the definition of the derivative we obtain the relation
If we set x1 = x + h and let h tend to zero by running through the sequence
h= 1/2, 1/3, 1/4,..., 1/n,...,
then, on applying the rules of logarithms, we find
By writing z = 1/x and using again the laws for the logarithm we obtain
In terms of the exponential function,
(12)
Here we have the famous formula defining the exponential function as a simple limit. In particular, for z = 1 we find
(13) e = lim (1 + 1/n)n,
and for z = –1,
(13a)
These expressions lead at once to expansions in the form of infinite series. By the binomial theorem we find that
or
It is plausible and not difficult to justify completely (the details are omitted here) that we can perform the passage to the limit as n → ∞ by replacing by 0 in each term. This gives the famous infinite series for ex,
(14)
and in particular the series for e,
which establishes the identity of e with the number defined on page 298. For x = –1 we obtain the series
which gives an excellent numerical approximation with very few terms, the total error involved in breaking off the series at the nth term being less than the magnitude of the (n + 1)st term.
By exploiting the differentiation formula for the exponential function we can obtain an interesting expression for the logarithm. We have
as h tends to 0, because this limit is the derivative of ey for y = 0, and this is equal to e0 = 1. In this formula we substitute for h the values z/n, where z is an arbitrary number and n ranges over the sequence of positive integers. This gives
or
as n tends to infinity. Writing z = log x or ez = x, we finally obtain
(15)
(see p. 323), this represents the logarithm as the limit of a product, one of whose factors tends to zero and the other to infinity.
Miscellaneous Examples and Exercises. By including the exponential function and the logarithm we now master a large class of functions and have access to many applications.
Differentiate: 1) x(log x –1). 2) log (log x). 3) . 4) . 5)e–x2. 6) eex (a compound function es with z = es). 7) xx (Hint: xx = ex log z). 8) log tan x. 9) log sin x; log cos x. 10) x/log x.
Find the maxima and minima of 11) xe–x, 12) x2e–x, 13) xe–ax.
*14) Find the locus of the maximum point of the curve y = xe–ax as a varies.
15) Show that all the successive derivatives of e–z2 have the form e–x2 multiplied by a polynomial in x.
*16) Show that the nth derivative of e–1/x2 has the form e–1/x2 multiplied by a polynomial of degree 2n – 2.
*17) Logarithmic differentiation. By using the fundamental property of the logarithm, the differentiation of products can sometimes be effected in a simplified manner. We have for a product of the form
p(x) = f1(x)f2(x)... fn(x),
D(log p(x)) = D(log f1(x)) + D(log f2(x)) +... + D(log fn(x)),
and hence, by the rule for differentiating compound functions,
Use this for differentiating
a) x(x + 1)(x + 2)... (x + n), b) xe–ax2.
5. Infinite Series for the Logarithm. Numerical Calculation
It is not formula (15) that serves as the basis for numerical calculation of the logarithm. A quite different and more useful explicit expression of great theoretical importance is far better suited to this purpose. We shall obtain this expression by the method used on page 441 for finding π, exploiting the definition of the logarithm by formula (1). One small preparatory step is needed; instead of aiming at log x, we shall try to express y = log (1 + x), composed of the functions y = log z and z = 1 + x. We have . Hence log(1 + x) is a primitive function of 1/(1 + x), and we infer by the fundamental theorem that the integral of 1/(1 + u) from 0 to x is equal to log (1 + x) – log 1 = log (1 + x); in symbols,
(16)
(Of course, this formula could just as well have been obtained intuitively from the geometrical interpretation of the logarithm as an area. Compare p. 413.)
In formula (16) we insert, as on page 442, the geometrical series for (1 + u)–1, writing
where, cautiously, we choose to write down not an infinite series, but rather a finite series with the remainder
Substituting this series in (16) we may use the rule that such a (finite) sum can be integrated term by term. The integral of us from 0 to x yields , and thus we obtain immediately
where the remainder Tn is given by
We shall now show that Tn tends to zero for increasing n provided that x is chosen greater than –1 and not greater than +1, in other words, for
-1 < x ≤ 1,
where it is to be noted that x = +1 is included, while x = –1 is not. According to our assumption, in the interval of integration u is greater than a number –α, which may be near to –1 but is at any rate greater than –1, so that 0 < 1 – α < 1 + u. Hence in the interval from 0 to x we have
and therefore
or
Since 1 – α is a fixed factor, we see that for increasing n this expression tends to 0 so that from
(17)
we obtain the infinite series
(18)
which is valid for –1 < x ≤ 1.
If, in particular, we choose x = 1, we obtain the interesting result
(19)
This formula has a structure similar to that of the series for π/4.
The series (18) is not a very practical means for finding numerical values for the logarithm, since its range is limited to values of 1 + x between 0 and 2, and since its convergence is so slow that one must include many terms before obtaining a reasonably accurate result. By the following device we can obtain a more convenient expression. Replacing x by –x in (18) we find
(20)
Subtracting (20) from (18) and using the fact that log a – log b = log a + log (1/b) = log (a/b), we obtain
(21)
Not only does this series converge much faster, but now the left side can express the logarithm of any positive number z, always has a solution x between –1 and +1. Thus, if we want to calculate log 3 we set and obtain
With only 6 terms, up to , we find the value
log 3 = 1.0986,
which is accurate to five digits.