3.6 The Logarithm and the Exponential Function
The systematic relations between the differential and the integral calculus lead naturally to a convenient method of approach to the exponential function and the logarithm. Although we have already investigated these functions in 1.3.4 and A1.3, we now define them afresh and redevelop their theory without making any use of our previous definition and the results based on it. We shall begin with the logarithm and then obtain the exponential function as its inverse.
3.6.1 Definition of the Logarithm. The Differentiation Formula: We have seen that, in general, indefinite integration of the power xn for integral indices n leads to a power of x. The only exception is the function 1/x, which does not appear as the derivative of any of the functions which we have dealt with so far. It is natural to assume that the indefinite integral of the function 1/x represents a new sort of function; thus, pursuing this idea, we will investigate the function
for x > 0. We call it the logarithm of x or, more accurately, the natural logarithm of x, and write it y=loge x or y = ln x. We have denoted the variable of integration by x, in order to avoid confusion with the upper limit x.
The choice of the number 1 as lower limit is arbitrary; however, it will soon prove its convenience.
During the following argument, it will appear that the logarithm, thus defined, is the same as the logarithm which we defined previously in an elementary way. But, as we once more emphasize, the results of the following study are independent of those obtained earlier.
Geometrically, our
logarithmic function means the area shown shaded Fig. 14; it is
bounded above by the rectangular hyperbola y=l/x , below by
the x-axis, and on the sides by the lines x=1 and x=x.
This area is to be reckoned positive, if x>1,
negative if x<1. For x = 1, the area vanishes,
whence we have log 1 = 0.
According to the above definition, the derivative of the logarithm is given by the formula
We emphasize specially that we assume throughout that the argument x is positive; the logarithm of 0 or of any negative value cannot be formed in accordance with the formula above, because the integrand 1/x becomes infinite when x = 0. On the other hand, if we choose some negative number, say -1, as the lower limit, we can form the integral with a negative upper limit x, i.e. we can consider the expression
Owing to the significance of the integral as the limit of a sum or as an area, we see that for x < 0
Hence we can, in
general, write the formula for indefinite integration as
Naturally, the logarithm can be represented by means of a graph. This graph, the logarithmic curve, is shown in Fig.15. We have already seen in 2.4.5 how to construct it.
3.6.2 The Addition Theorem: The logarithm defined as above obeys the fundamental law:
The proof of this addition theorem follows directly from the differentiation formula. In fact, writing z=log (ax) and applying the chain rule, we obtain
However,
and, since the functions z and log x have the same derivative, they differ only by a constant, so that z = log x + c or
This being true for all positive values of x,
we first put x = 1 to find c; since
log 1 = 0, this yields
Substituting this value for c, we have
whence, for x = b,
which was to be proved.
For arbitrary positive numbers a1, a2, ··· , an, the equation
follows from the addition theorem for the logarithm.
In particular, if all the numbers a1, a2, ··· , an are equal to one and the same number a, we have
Similarly, it follows that
so that
Moreover, if we set 
, it follows that log a
= n log a, or
Hence, by repeated use of the addition theorem, we find that, when m is a positive integer,
The equation
is thus proved for all positive rational values of r, and it is obviously correct for r = 0. It is also valid for negative rational values of r, because then
3.6.3 Monotonic Character and Values of theLogarithm: Obviously, the value of the logarithm increases and decreases with x, whence the logarithm is a monotonic function.
Since the derivative l/x becomes smaller and smaller as x increases, the function increases more and more slowly as x increases. Nevertheless, as x increases beyond all bounds, the function log x does not tend to a positive limit, but becomes infinite, that is, for every positive number A, no matter how large, there are values of x for which log x > A. This fact follows very readily from the addition theorem. In fact, log 2n = n log2, and since log 2 is a positive number, we can, by taking x = 2n with sufficiently large values of n. make log x as large as we please.
Since log (1/2n) = - n log 2, we see that, as x tends to zero through positive values, log x is negative and increases numerically beyond all bounds.
In summary, we have: The function log x is a monotonic function which assumes all values between -¥ and +¥ as the independent variable x ranges over the continuum of positive numbers.
3.6.4 The Inverse Function of the Logarithm (the Exponential Function): Since the function y=log x (x > 0) is a monotonic function of x which assumes all real values, its inverse function, which we shall at first denote by x=E(y), must be a single-valued monotonic function defined for every real value of y; it is differentiable, since log x itself is differentiable. We interchange the notation for the dependent and independent variables and study the function E(x) in detail. In the first place, obviously, it must be positive for every value of x. Moreover, we must have
In fact, this equation is equivalent to the statement that log 1 = 0.
From the addition theorem for the logarithm follows immediately the multiplication theorem
In order to prove this, we merely need note that the equations
are equivalent to
Since, by the addition theorem for the logarithm, a + b = log ab, it must be true that c = ab, which proves the multiplication theorem.
We derive from this theorem a fundamental property of the function y = E(x), which gives us the right to call our function the exponential function and to write it symbolically in the form
In order to obtain this property, we observe that there must be a number - which we shall call e - for which
Its identity with the number e considered in in 1.6.5 will be proved in 3.6.6.
This is equivalent to the definition
Using the multiplication theorem for the function E(x), we have
and, in the same way, for positive integers m and n,
which we could also have found directly from the addition theorem for the logarithm.
The equation E(r) = er thus proved for positive rational numbers r holds also for negative rational numbers by virtue of the equation
The function E(x) is therefore a
function which is continuous for all values of x and for rational values of
x coincides with ex. These
facts give us the right to call our function ex
also for arbitrary irrational values of x . (It
should be noted that here the continuity of ex
is an immediate consequence of its definition as the inverse
function of a continuous, monotonic function, 
while, if the elementary definition is adopted, the
continuity must be proved.)
The exponential function is differentiated according to the formula
This formula expresses the important fact that the derivative of the exponential function is the function itself.
The proof is very simple. In fact, we have x = log y, whence, by the formula for the differentiation of the logarithm, we have dx/dy = 1/y, and then, by the rule for inverse functions,
as stated.
The graph of the exponential function ex - the so-called exponential curve - is obtained by reflection of the logarithmic curve in the line which bisects the first quadrant (Fig. 16).
3.6.5 The General Exponential Function ax and the General Power xa:
The exponential Function ax for an arbitrary positive base a is now simply defined by the equation
which agrees with the earlier definition by virtue of the relation
By the chain rule, we immediately obtain
The inverse function of the exponential function y = ax is called the logarithm to the base a and is written
while the logarithmic function previously introduced, when it is required to distinguish it, is spoken of as the natural logarithm, or the logarithm to the base e.
It follows immediately from the definition that
which shows that the logarithm of y for an arbitrary positive base a ¹ 1 is obtained by multiplying the natural logarithm of y by the reciprocal of the natural logarithm of a, the modulus of the system of logarithms to the base a.
If we take a = 10, we obtain the ordinary Briggian logarithms, which have already been encountered in elementary mathematics and which are advantageous for use in numerical computations.
Instead of our previous definition of the general power xa' (x > 0), we shall now define this power by the equation
The rule for differentiating the power xa follows immediately from the definition, using the chain rule; for
in agreement with our previous result .
3.6.6. The Exponential Function and the Logarithm represented as Limits: We are now in a position to state important limiting relations referring to the quantities introduced above. We begin with the formula for differentiating the function f(x) = log x:
If we set 1/x = z, this becomes
Since the function ex is continuous for all values of x, this implies that
In particular, if we give h the sequence of values 1, 1/2, 1/3, ··· , 1/n, ··· , we have
If we assign to z the value 1, Equation (a) states the important fact:
As h tends to zero, the expression (1 + h)1/h tends to the number e:
Equation (b) yields
which proves that the number e is the same as the number denoted earlier by the symbol e.
It follows from the differentiation formula for ax
that for x = 0
which expresses the logarithm of a directly as a limit.
We append to this equation the remark that we can complete by it the earlier obtained relation
We have always been obliged to exclude the case a = -l. However, we can now discover what happens when the number a tends to the limit -1. If we put a=1, the left-hand side will, by our definition of the logarithm, have the limit
whence the right-hand side has the same limit when a ® -1.
We have here carried out the passage to the limit ® -1 under the integral sign without further investigation; cf. the earlier discussion.
Moreover, this fact agrees with the formula
we only need write a + 1 = h.
We have thus cleared up the exceptional case a = -1 in the integration formula which we have so often used. The formula above is still meaninglesS when a= -1, but it retains as a limit formula its significance as a ® -1.
3.6.7 Final Remarks: We review here briefly the train of thought followed in this section, We have first defined the natural logarithm y = log x as for x > 0 by means of an integral, whence we immediately deduced the differentiation formula, the addition theorem, and the existence of an inverse. We then investigated the inverse function y = ex, where the number e was seen to be the number the logarithm of which is 1, and derived its differentiation formula as well as limit expressions for it and for the logarithm. The introduction of the functions y = xa = ea log x and y = ax = ex log a followed naturally.
In the discussion given here, in contrast to the elementary treatment, the question of continuity causes no difficulty, since the logarithm is defined as an integral and therefore as a continuous and differentiable function, the inverse function of which is also continuous.
1. Sketch the function y = 1/x (1 £ x £ 2) on a large scale, using graph paper, and find loge2 by counting squares.
Differentiate the functions:
6. Differentiate 
(a) by using
the chain rule and the quotient rule, without preliminary
simplification, (b) first simplifying by means of the
theorems on logarithms.
7. (a) Differentiate 
,
(b) Differentiate the same function, first taling
logarithms and simplifying it.
8.* Given 
, prove that 
9. Show that the function 
satisfies the equation
for all values of a and b.
10.* Show that 
when x ¹ 0, where Pn(x)
is a polynomial of degree 2n -2. Establish the recurrence
relation
11. Find the maximum of 
where l and a are constants. Find the locus of this maximum when l is
allowed to vary.
12. Differentiate 
13. Differentiate asin x(log x)·sin x(log x).
3.7 Some Applications of the Exponential Function
We shall consider now some different problems involving the exponential function and thus gain insight into the fundamental importance of this function for all kinds of applications.
3.7.1 Definition of the Exponential Function by Means of a Differential Equation: We can define the exponential function by a simple theorem the use of which will save us many detailed investigations of particular cases:
If a function y = f(x) satisfies an equation of the form
where a is a constant other than zero, then y has the form
where c if also a constant; conversely, every function of the form ceax satisfies the equation y' = ay.
The last equation is usually briefly referred to as a differential equation, since it expresses a relation between the function and its derivative. In order to make the theorem clear, we note first of all that in the simplest case a =1 the above equation becomes y' = y. We know that y = ex satisfies this equation and it is clear that the same is also true of y = cex, where c is an arbitrary constant. Conversely, we can easily see that no other function satisfies the differential equation. For if y is such a function, we consider the function u = ye-x. We must then have
But the right-hand side vanishes, since we have assumed that y' = y, whence u' = 0, so that, by 2.4.3, u is a constant c and y = cex, as we wished to prove.
The case of any non-zero value of a can be treated in exactly the same manner as the special case a = 1. If we introduce the function u = ye-ax, we obtain the equation u' = y'e-ax — ae-ax. Hence we find from the assumed differential equation that u' = 0, so that u = c and y = ceax. The converse is clear.
We will now apply this theorem to a number of examples and thereby make it more intelligible.
3.7.2 Interest, Compounded Continuously. Radio-active Disintegration: A capital sum, or principal, which has its interest added to it at regular periods of time, increases by jumps at these interest periods in the following manner. If 100a is the rate of interest per cent and moreover the interest accrued is added to the principal at the end of each year, then after x years the accumulated amount of an original principal of 1 will be
However, if the principal had the interest added to it not at the end of each year, but at the end of each n-th part of a year, then after x years the principal would amount to
Taking x = 1, for the sake of simplicity, i.e, reckoning the interest at 100a per cent for one year, we find that, if the interest is computed in this latter way, the principal 1 amounts after one year to
If we now let n increase beyond all bounds, i.e., if we let the interest be calculated at shorter and shorter intervals, the limiting case will signify in a sense that the interest is compounded continuously, at each instant; and we see that the total amount after one year will be ea times the original principal. Similarly, if the interest is calculated in this manner, an original principal of 1 will have grown after x years to an amount eax, where x may be any number, integral or otherwise.
The discussion in 3.7.1 forms a framework within which examples of this type are readily understood. We consider a quantity y, which increases (or decreases) with time. Let the rate at which this quantity changes be proportional to the total quantity. Then, if we take time as the independent variable x, we obtain a law of the form y' = a y for the rate of increase, where a, the factor of proportionality, is positive or negative, according to as the quantity is increasing or decreasing, Then, in accordance with 3.7.1, the quantity y itself will be given by a formula
where the meaning of the constant c is immediately obvious, if we consider the instant x = 0. At that instant, ea x = 1 and we find that c = y0 is the. quantity at the beginning of the time under consideration, so that we may write
A characteristic example of the use of these ideas is the case of radioactive disintegration. The rate at which the total quantity y of a radioactive substance is diminishing at any instant is proportional to the total quantity present at that instant; this is a priori plausible, as each portion of the substance decreases as rapidly as every other portion. Hence the quantity y of the substance, expressed as a function of the time, satisfies a relation of the form y ' = - ky, where k is to be taken as positive, since we are dealing with a diminishing quantity. The quantity of substance is thus expressed as a function of the time by y = 3.8y0e-kx , where y0 is the amount of the substance at the beginning of the time (time x = 0).
After a certain time t, the radioactive substance will be diminished to half its original quantity. This so-called half-period is given by the equation
whence we immediately obtain 
3.7.3 Cooling or Heating of a Body by a Surrounding Medium: Another typical example of the occurrence of the exponential function is offered by the cooling of a body, e.g., a metal plate, which is immersed in a very large bath of a given temperature. In considering this cooling, we assume that the surrounding bath is so large that its temperature is unaffected by the cooling process. Moreover, we assume that at each instant all parts of the immersed body are at the same temperature and that the rate at which the temperature changes is proportional to the difference between the temperature of the body and that of the surrounding medium (Newton's law of cooling).
If we denote the time by x and the temperature difference by y = y(x), this law of cooling is expressed by the equation
where k is a positive constant the value of which depends on the body itself. From this instantaneous relationship, which expresses the effect of the cooling process at a given instant, we now wish to derive an integral law which will allow us to find the temperature at an arbitrary time x from the temperature at an initial time x = 0. The theorem of 3.7.1 immediately gives us this integral law in the form
where k is the above-mentioned constant depending on the body. This shows that the temperature decreases exponentially and tends to become equal to the external temperature. The rapidity with which this happens is expressed by the number k. As before, we find the meaning of the constant c by considering the instant x = 0; this yields y0 = c, so that our law of cooling can finally be written is the form
Obviously, the same discussion will also apply to the heating of a body. The only difference is that the initial difference of the temperature y0 is in this case negative instead of positive.
3.7.4 Variation of the Atmospheric Pressure with the Height above the Surface of the Earth: As another example of the occurrence of the exponential formula we shall derive the law according to which the atmospheric pressure varies with the height. We use here: (1) the physical fact that the atmospheric pressure is equal to the weight of the column of air vertically above a surface of area 1 and (2) Boyle's law, according to which the pressure of the air (p) at a given constant temperature is proportional to the density of the air (s). Boyle's law, expressed in symbols, is p = as, where a is a constant which depends on a specific physical property of the air and, in addition, it is proportional to the absolute temperature—here, we are not concerned with this, as we shall assume that the temperature is constant. Our problem is to determine p = f(h) as a function of the height (h) above the surface of the Earth.
If we denote by p0 the atmospheric
pressure at the surface of the Earth, i.e., the total weight of
the air column supported by unit area, and by s(l) the density of the air at the
height l above the Earth, the
weight of the column up to the height h will he given by
the integral 
. Hence the pressure at height h will
be
By differentiation, this yields the relationship between the pressure p = f(h) and the density s(h):
We now use Boyle's law to eliminate the quantity s from this equation and obtain an equation
which only involves the unknown pressure function. It follows from 3.7.1 that
If we denote, as above, the pressure at the Earth's surface, i.e., f(0), by p0, it follows immediately that c = p0, and consequently
These two formulae find frequent application. For example, if the constant a is known, they enable us to find the height of a place from the barometric pressure or to find the difference of the heights of two locations by measuring the local atmospheric pressures. Again, if the atmospheric pressure and the height h are known, we can determine the constant a, which is of great importance in gas theory.
3.7.5 Progress of a Chemical Solution: We will now consider an example from chemistry, namely the so-called molecular reaction. Let a substance be dissolved in a relatively large amount of solvent, say a quantity of cane sugar in water. If a chemical reaction takes place, the chemical law of mass action in this simple case states that the rate of reaction is proportional to the quantity of reacting substance present. If we assume that the cane-sugar is being transformed by catalytic action into invert sugar and if we denote by u(x) the quantity of cane sugar which at time x is still unchanged, the velocity of reaction will be -du/dx, and, in accordance with the law of mass action, there holds an equation of the form
where k is a constant which depends on the reacting substance. From this instantaneous law, we immediately obtain, as in 3.7.1, an integral law, which gives us the amount of cane sugar as a function of the time
This formula show us clearly how the chemical reaction tends asymptotically to its final state u = 0, that is, the complete transformation of the reacting substance. The constant a is obviously the quantity present at x = 0.
3.7.6 Making and Breaking of an Electric Circuit: As a final example, consider the growth of a (direct) electric current when a circuit is completed (or its decay, when it is broken). If R is the resistance of the circuit and I the impressed electro-motive force (Voltage), the current I will gradually increase from its original value 0 to the steady final value E/R. Hence we must consider I as a function of the time. The growth of the current depends on the self-induction of the circuit ; the circuit has a characteristic constant L - the coefficient of self-induction - of such a nature that as the current increases, an electro-motive force of magnitude LdI/dx, opposed to the external electromotive force E, is developed. We obtain from Ohm's law, according to which the product of the resistance and the current is at each instant equal to the actual effective voltage, the relation
Hence, we put
we find immediately that f '(x) = - R/L· f(x), so that, by the theorem in 3.7.1, f(x)=f(0)e-Rx/L . Recalling that I(0) = 0, we see that f(0) = -E/R and thus obtain the expression for the current as a function of the time
We see from this expression that as the circuit is closed the current tends asymptotically to its steady value E/R.
1. The function f(x) satisfies the equation
(a) If f(x) is differentiable,
either f(x) º 0
or else f(x) = eax,
(b)* Iff(x) is continuous, either f(x)
º 0 or
else f(x) = eax.
2. If a differentiable function f(x) satisfies the equation
then f(x) = a log x.
3. A quantity of radium weighs 1 g at time t = 0, At time t = 10 (years), it has diminished to 0.997g. After what time will it have diminished to 0.5 g?
4. Solve the differential equations:
3.8.1 Analytical Definition: In many applications, the exponential function does not enter by itself, but in combinations of the form
It is convenient to introduce these and similar combinations as special functions with the notation:
and call them the hyperbolic sine, hyperbolic cosine, hyperbolic 
tangent, and hyperbolic cotangent,
respectively. The
functions sinh x, cosh x
and tanh x are defined for all values of x, while
in the case of coth x the point x= 0 must be
excluded. This notation is designed to express a certain analogy
with the trigonometric functions; it is this analogy, which we
are about to study in detail. as it justifies special
consideration of these new functions. Figs. 17, 18 and 19 show
the graphs of the hyperbolic functions; the dotted lines in
Fig.17 are the graphs of y = ex/2
and y = e-x/2,
from which the graphs of sinh x and cosh x may
readily be constructed.We see that cosh x is an even function, i.e., a function which remains unchanged when
x is replaced by -x, while sinh x is an odd function, i.e., a function which changes sign when x is
replaced by -ii.
The function
is (by its definition) positive for all values of x. It has its minimum when x = 0, while cosh 0 = 1.
There exists
between cosh x and sinh x the fundamental relation
which follows immediately from the definitions of these functions.If we now denote the independent variable by t instead of x and write
we have
thus, the point with the coordinate x=cosh
t, y = sinh t moves along the
rectangular hyperbola x² - y² 
=1 as t runs through the whole scale of values
from -¥ to +¥. According to the defining equation, x ³ 1 and we may
easily convince ourselves that y runs through the whole scale of
values -¥ to +¥ with t; in fact, if t tends to infinity so
does et, while e-r
tends to zero. We may therefore state more exactly that as t
runs from -¥ to +¥, the equations x = cosh t, y =
sinh t give us one branch, namely, the right-hand one,
of the rectangular hyperbola.
3.8.2 Addition Theorems and Formulae for Differentiation: From the definitions of our functions follow the formulae, known as addition theorems:
The proofs are obtained at once by writing
and setting in these equations
The analogy between these formulae and the corresponding trigonometrical formulae is clear. The only difference in the addition theorems is one sign in the first formula.
A corresponding analogy holds for the differentiation formulae. Remembering that d(ex)/dx = ex, we readily find
At times, it is convenient to introduce the functions sech x = 1/cosh x, cosech x = 1/sinh x.
3.8.3 The Inverse Hyperbolic Functions: To the hyperbolic functions x = cosh t, y = sinh t correspond the inverse functions
The notation cosh-1x, etc. is also used.
Since the function sinh t is monotonic increasing throughout the interval - ¥ < t < ¥, its inverse function is uniquely determined for all values of y; on the other hand, we learn from a glance at the graph (Fig. 17) that t = ar cosh x is not uniquely determined, but has an ambiguity of sign, because there corresponds to a given value of x not only the number t, but also the number -t. Since cosh t³1 for all values of t, its inverse arcosh x is defined only for x³1.
We can express these inverse functions very easily in terms of the logarithm by regarding the quantity et = u in the definitions
as unknowns and solving these (quadratic) equations for u. Then
since u = et can have only positive values, the square root in the second equation must be taken with the positive sign, while in the first equation either sign is possible. In the logarithmic form
In the case of arcosh x, the variable x is restricted to the interval x ³ 1, while arsinh y is defined for all values of y.
The formula gives us two values, 
and 
for
arcosh x, corresponding to the two branches of arcosh x.
Since
the sum of these two values of arcosh x is zero, which agrees with an earlier remark.
The inverses of the hyperbolic tangent and hyperbolic cotangent can be defined analogously and can also be expressed in terms of logarithms. We denote these functions by artanh x and arcotanh x; expressing the independent variable everywhere by x, we readily obtain
The differentiation of these inverse functions should be carried out by the reader himself; he may use either the rule for differentiating an inverse function or the chain rule in conjunction with the above expressions for the inverse functions in terms of logarithms. If x is the independent variable, the results are
The last two formulae do not contradict each
other, since the first holds only for -1 < x < 1
and the second only for x < -1 and 1 < x. The
two values of
dar cosh x/dx, expressed by the sign ± in
the first formula, correspond to the two different branches of
the curve
3.8.4 Further Analogies: In the above representation of the rectangular hyperbola by the quantity t, we did not attempt to bring out any geometrical meaning of the parameter t itself. We shall now return to this matter and thus gain still more insight into the analogy between the trigonometric and hyperbolic functions. If we represent the circle by the equation x² + y² = 1 by means of a parameter t in the form x = cos t, y = sin t, we can interpret the quantity t as an angle or a length of arc measured along the circumference; however, we may also regard f as twice the area of the circular sector, corresponding to that angle, the area being reckoned positive or negative a«cording to as the angle is positive or negative.
We now make the analogous statement that for the hyperbolic functions the quantity t is twice the area of the hyperbolic sector, shown shaded in Fig. 20.
Just as the notation t = arcos x recalls that t is an arc of a circle of reference, so t = ar cosh x recalls that t is a certain area connected with a rectangular hyperbola.
The proof is obtained without difficulty, if we refer the hyperbola to its asymptotes as axes by means of the coordinate transformation
or
with these new coordinates, the equation of the hyperbola becomes xh = ½. We thus see immediately that the area in question is equal to the area of the figure ABQP; in fact, the two right-angled triangles OPQ and OAB have the same area, according to the equation of the hyperbola. ' The two points A and P obviously have the coordinates
respectively, and we thus obtain for double the area of our figure
A comparison of this expression with the formula in 3.8.3 for the inverse function t = ar cosh x shows us that our statement about the quantity t is true.
In conclusion, it may be pointed out that, as shown in Fig. 21 above, the hyperbolic functions can be diagrammatically represented on the circle.
Numerical values of the hyperbolic functions, which are useful in a variety of calculations, can be found in many tables,some of which are: J. B. Dale, Five figure Tables of Mathematical Functions (Ainold, 1918}, K. Hayashi, Fünfstellige Tafeln der Kreis- und Hyperbel-funktionen (Berlin, 1930); E. Jahnlie and P. Emde, Funktionentafeln mit Formeln und Kurven (German and English, Leipzig 1933.
Exercises:
1. Prove the formula
Obtain similar formulae for sinh a - sinh b, cosh a + cosh b, cosh a — cosh b.
2. Express tanh(a ± b) in terms of tanh a and tanh b, coth(a ± b) in terms of coth a and coth b, sinh a/2 and cosh a/2 in terms of cosh a.
3. Differentiate
4. Calculate the area bounded by the catenary y = cosh x, the ordinates x = a and x = b, and the x.axis.
3.9 The Order of Magnitude of Functions
The various functions which we have met in this chapter exhibit very important differences as regards their behaviour for large values of their arguments or, as we also say, of the order of magnitude of their increase. In view of the great importance of this, we shall discuss here briefly this matter, even though it is not directly linked to the idea of the integral or of the derivative.
3.9.1 The Concept of Order of Magnitude. The simplest Cases: If the variable x increases beyond all bounds, then, for a > 0, the functions a, log x, ex, eax will also increase beyond all bounds. However, as regards the manner of this increase, we can immediately point out an essential difference between them. For example, the function x³ will become infinite to a higher order than x²; this means that, as x increases, the quotient x³/x² itself increases beyond all bounds. Similarly, we shall say that the function xa becomes infinite to a higher order than xb , if a > b > 0, etc.
Quite generally, we shall say of
two functions f(x) and g(x), the
absolute values of which increase with x beyond all
bounds, that f(x)
becomes infinite of a higher order than g(x), if, as x increases, the quotient |f(x)/g(x)|
increases beyond all bounds; we shall say that f(x)
becomes infinite of a lower order than g(x), if the quotient |f(x)/g(x)|
tends to zero as x increases; and we shall say that the
two functions become infinite of the same order of magnitude if
as x increases the quotient |f(x)/g(x)|
has a limit other than 0 or at least remains between two
fixed positive bounds. For example, the function ax³+bx²+c=f(x),
where a ¹ 0, will be of the same order of magnitude as the
function x³=g(x); in fact, the quotient 
has the
limit |a|. On the other hand, the function x³ + x
+ 1 becomes infinite of a higher order of magnitude than the
function x² + x + 1.
A sum of two functions f(x) and f(x), where f(x) is of higher order of magnitude than f(x), has the same order of magnitude as f(x). In fact,
and, by hypothesis, this expression tends to 1as x increases.
We might be tempted to measure the order of magnitude of functions by a scale, assigning to the quantity x the order of magnitude 1 and to the power xa (a > 0) the order of magnitude a. Obviously, then a polynomial of the n-th degree has the order of magnitude n; a rational function, the degree of the numerator of which is greater than that of the denominator by h, has then the order of magnitude h.
3.9.2 The Order of Magnitude of the Exponential Function and the Logarithm: However, it turns out that any attempt to fix the order of magnitude of an arbitrary function by the above scale must end in failure. In fact, there are functions which become infinite of higher order than the power xa of x, no matter however large is the chosen a ; again, there are functions which become infinite of lower order than the power xa, no matter how small the positive number a is chosen. Hence, these functions will not fit in anywhere into our scale. Without entering into a detailed theory of order of magnitude we shall prove the theorem: If a is an arbitrary number greater than 1, then the quotient ax tends to infinity as x increases. In order to prove this statement, we construct the function
obviously, it is sufficient to show that this increases beyond all bounds if x tends to +¥. For this purpose, we consider the derivative
and note that for x ³ 0 this is not less than the positive number 1/2 log a, whence, for x ³ c
and the right-hand side becomes infinite as x increases.
We shall give a second proof of this important
theorem. If we write 
we have b > 1 and h
> 0. Let n be an integer such that n£x <n+1; we may take x
> 1, so that n ³ 1. Applying the Lemma of l.5.3, we have
so that
whence it tends to infinity with x.
There follow many results from the fact just proved. For example, for every positive index a and every number a > 1, the quotient ax/xa tends to infinity as x increases, that is: The exponential function becomes infinite of a higher order of magnitude than any power of x.
In order to see this, we need only show that the a-th root of the expression, that is,
tends to infinity. However, this follows immediately from the preceding theorem, when x is replaced by y = x/a.
We can prove in a similar fashion the theorem: For every positive value of a, the quotient (log x)/xa tends to zero when x tends to infinity, that is, the logarithm becomes infinite at a lower order of magnitude than any arbitrarily small positive power of x.
The proof follows immediately, if we set log x = y, so that our quotient is transformed into y/ea y. We then set ea = a; then a is a number > 1 and our quotient y/ay approaches 0 as y increases. Since y approaches infinity with x, our theorem is proved.
Another very simple proof maybe suggested: For x > 1 and e > 0,
if we choose e smaller than a and divide both members of the inequality by xa, then as x ® ¥ it follow that (log x)/xa ® 0.
Based on these results, we can construct
functions of an order of magnitude by far higher than that of the
exponential function and other functions of an order of magnitude
far lower than that of the logarithm. For example, the function 
is of higher order than the
exponential function, and the function log log x of
lower order than the logarithm; obviously, we can repeat these
iteration processes as often as we like, piling up the symbols e
or log to any extent we please.
3.9.3 General Remarks: These considerations show that it is not possible to
assign by systematic reasoning to all functions definite numbers
as orders of magnitude in such a way that, when two functions are
compared, the function of the higher order of magnitude has the
higher number. For example, if the function x is of the
order of magnitude 1 and the function x1+e
of the order of magnitude 1 + e, then the function x log x must
be of an order of magnitude which is greater than I and less than
1 + e no matter how small e is chosen. But there exists no such number. Moreover,
it is readily seen that functions need not possess a clearly
defined order of magnitude. For example, the function 
approaches no definite limits as x increase; on the
contrary, for x = np (where n is
an integer) its value is 1/np, while, for x =
(n + 1/2)p, it is (n + 1/2)p +1+
1/(n + ½)p. Although both the numerator and denominator become
infinite, the quotient neither remains between positive bounds,
nor tends to zero, nor tends to infinity. Hence, the numerator is
neither of the same order as the denominator, nor of lower order,
nor of higher order. This apparently startling situation merely
means that our definitions are not designed in such a way that we
can compare any pair of functions. This is not a defect; we have
no desire to compare the orders of such functions like the
numerator and denominator above, since knowledge of the value of
one of them gives us no useful information about the other one.
3.9.4 The Order of Magnitude of a Function in the Neighbourhood of an Arbitrary Point: Just as we can investigate the behaviour of a function when x increases without limit, we may also ask ourselves whether and how functions which become infinite at the point x = x may be distinguished with regard to their behaviour at that point. Moreover, we state that the function f(x)= 1/|x - x| becomes infinite at the first order at the point x = x, and correspondingly that the function 1/|x - x|a becomes infinite at the order a, provided that a is positive.
We then recognize that the function e1/|x - x| becomes infinite at higher order, and the function log |x - x| infinite at lower order than all these powers, i.e., that there hold the limiting relations
In order to see this, we merely set 1/|x - x| = y; our statements then reduce to the known theorem, since
and y increases beyond all bounds as x tends to x· The method of reduction of the behaviour at a finite point to the behaviour at infinity by the substitution 1/|x-x| = y frequently proves useful.
3.9.5 The Order of Magnitude of a Function tending to Zero: Just as we seek to describe the approach of a function to infinity more definitely by means of the concept of order of magnitude, we may also specify the way in which a function approaches zero. We say that, as x ® ¥, the quantity 1/x vanishes to the first order, the quantity x -a, where a is positive, to the order a. We find once again that the function 1/log x vanishes to a lower order than an arbitrary power x -a, that is, the relation
applies for every positive a.
In the same way, we say that, for x = x, the quantity x - x vanishes to the first order, the quantity |x-x|a to the order a. With the above results, it is easy to prove the relations
which are usually expressed as follows:
The function 1/log |x| vanishes as x ® 0 to a lower order than any power of x; the exponential function e-1/|x| vanishes to a higher order than any power x.
1. Compare the following functions with powers of x as regards their order of magnitude as x®¥:
2. Compare the functions of Exercise 1. with 
.
3. Compare the functions of Exercise 1. with powers of x as x ® 0.
4. Does the 
exist?
6. What are the limit as x®¥ , of 
?
6. Let f(x) be a continuous function vanishing, together with its first derivative, for x = 0. Show that f(x) vanishes to a higher order than x as x®0.
7. Show that
when a0, b0 ¹ 0, is of the same order of magnitude as xm-n, when x®¥.
8.* Prove that ex is not a rational function.
9.* Prove that ex cannot satisfy an algebraic equation with polynomials in x as coefficients.