The Riemann Integral
5.4 The logarithm and the exponential
5.4.2 The exponential
Just as with the logarithm we define the exponential via a list of properties. Proposition 5.4.2. There exists a unique function E: R→(0,∞)such that
(i) E(0) =1.
(ii) E is differentiable and E0(x) =E(x). (iii) E is strictly increasing, bijective, and
lim
x→−∞E(x) =0, and xlim→∞E(x) =∞. (iv) E(x+y) =E(x)E(y)for all x,y∈R.
(v) If q∈Q, then E(qx) =E(x)q.
Proof. Again, we prove existence of such a function by defining a candidate, and prove that it satisfies all the properties. TheLdefined above is invertible. LetE be the inverse function ofL. Property is immediate.
Property follows via the inverse function theorem, in particular : Lsatisfies all the hypotheses of the lemma, and hence
E0(x) = 1
L0 E(x) =E(x).
Let us look at property . The functionE is strictly increasing sinceE0(x) =E(x)>0. AsE is the inverse ofL, it must also be bijective. To find the limits, we use thatE is strictly increasing and onto(0,∞). For everyM>0, there is anx0such thatE(x0) =MandE(x)≥Mfor allx≥x0. Similarly, for everyε >0, there is anx0such thatE(x0) =ε andE(x)<ε for allx<x0. Therefore,
lim
n→−∞E(x) =0, and nlim→∞E(x) =∞.
To prove property we use the corresponding property for the logarithm. Takex,y∈R. AsL
is bijective, findaandbsuch thatx=L(a)andy=L(b). Then E(x+y) =E L(a) +L(b)
=E L(ab)
=ab=E(x)E(y).
Property also follows from the corresponding property ofL. Givenx∈R, letabe such that
x=L(a)and
E(qx) =E qL(a)E L(aq
)
=aq=E(x)q.
Finally, uniqueness follows from and . LetE andF be two functions satisfying and . d
dx
F(x)E(−x)=F0(x)E(−x)−E0(−x)F(x) =F(x)E(−x)−E(−x)F(x) =0.
Therefore, by ,F(x)E(−x) =F(0)E(−0) =1 for all x∈R. Next, 1=E(0) =
E(x−x) =E(x)E(−x). Then
0=1−1=F(x)E(−x)−E(x)E(−x) = F(x)−E(x)
Finally,E(−x)6=0 for allx∈R. SoF(x)−E(x) =0 for allx, and we are done.
Having provedE is unique, we define theexponentialfunction as exp(x):=E(x).
Ify∈Qandx>0, then
xy=exp ln(xy)
=exp yln(x) .
We can now make sense of exponentiationxyfor arbitraryy∈
R; ifx>0 andyis irrational, define
xy:=exp yln(x) .
As exp is continuous thenxy is a continuous function ofy. Therefore, we would obtain the same result had we taken a sequence of rational numbers{yn}approachingyand definedxy=limxyn.
Define the numbere, sometimes calledEuler’s numberor thebase of the natural logarithm, as e:=exp(1).
Let us justify the notationexfor exp(x):
ex=exp xln(e)
=exp(x).
Let us extend properties of logarithm and exponential to irrational powers. The proof is immediate.
Proposition 5.4.3. Let x,y∈R.
(i) exp(xy) = exp(x)y
.
(ii) If x>0thenln(xy) =yln(x).
Remark 5.4.4. There are other equivalent ways to define the exponential and the logarithm. A common way is to defineE as the solution to the differential equationE0(x) =E(x),E(0) =1. See , for a sketch of that approach. Yet another approach is to define the exponential function by power series, see .
Remark5.4.5. We have proved the uniqueness of the functions LandE from just the properties L(1) =0,L0(x) =1/x and the equivalent condition for the exponential E0(x) =E(x), E(0) =1. Existence in fact also follows from just these properties. Alternatively, uniqueness also follows from the laws of exponents, see the exercises.
∗Eis a function into(0,∞)after all. However,E(
−x)6=0 also follows fromE(x)E(−x) =1. Therefore, we can prove uniqueness ofEgiven and , even for functionsE:R→R.
5.4.3
Exercises
Exercise5.4.1: Let y be any real number and b>0. Define f: (0,∞)→Rand g:R→Ras, f(x):=xy
and g(x):=bx. Show that f and g are differentiable and find their derivative.
Exercise5.4.2: Let b>0, b6=1be given.
a) Show that for every y>0, there exists a unique number x such that y=bx. Define thelogarithm baseb,
logb: (0,∞)→R, bylogb(y):=x.
b) Show thatlogb(x) = lnln((xb)).
c) Prove that if c>0, c6=1, thenlogb(x) =logc(x)
logc(b).
d) Provelogb(xy) =logb(x) +logb(y), andlogb(xy) =ylogb(x).
Exercise5.4.3(requires ): Use to study the remainder term and show that for all x∈R
ex= ∞
∑
n=0 xn n!.Hint: Do not differentiate the series term by term (unless you would prove that it works).
Exercise5.4.4: Use the geometric sum formula to show (for t6=−1)
1−t+t2− ···+ (−1)ntn= 1
1+t−
(−1)n+1tn+1
1+t .
Using this fact show
ln(1+x) = ∞
∑
n=1 (−1)n+1xn nfor all x∈(−1,1](note that x=1is included). Finally, find the limit of the alternating harmonic series
∞
∑
n=1 (−1)n+1 n =1−1/2+1/3−1/4+··· Exercise5.4.5: Show ex=nlim →∞ 1+x n n .Hint: Take the logarithm. Note: The expression 1+xnn
arises in compound interest calculations. It is the amount of money in a bank account after 1 year if 1 dollar was deposited initially at interest x and the interest was compounded n times during the year. The exponential exis the result of continuous compounding.
Exercise5.4.6:
a) Prove that for n∈Nwe have
n
∑
k=2 1 k ≤ln(n)≤ n−1∑
k=1 1 k. b) Prove that the limitγ:= lim n→∞ n
∑
k=1 1 k−ln(n) !exists. This constant is known as theEuler–Mascheroni constant . It is not known if this constant is rational or not. It is approximatelyγ≈0.5772.
∗Named for the Swiss mathematician (1707–1783) and the Italian mathematician (1750–1800).
Exercise5.4.7: Show
lim
x→∞
ln(x)
x =0.
Exercise5.4.8: Show that exisconvex, in other words, show that if a≤x≤b then ex≤ea b−x
b−a+eb xb−−aa.
Exercise5.4.9: Using the logarithm find
lim
n→∞n 1/n.
Exercise5.4.10: Show that E(x) =exis the unique continuous function such that E(x+y) =E(x)E(y)and
E(1) =e. Similarly, prove that L(x) =ln(x)is the unique continuous function defined on positive x such that
L(xy) =L(x) +L(y)and L(e) =1.
Exercise 5.4.11(requires ): Since(ex)0 =ex, it is easy to see that ex is infinitely differentiable (has
derivatives of all orders). Define the function f:R→R.
f(x):=
(
e−1/x if x>0,
0 if x≤0.
a) Prove that for any m∈N,
lim
x→0+
e−1/x
xm =0.
b) Prove that f is infinitely differentiable.
c) Compute the Taylor series for f at the origin, that is,
∞
∑
k=0 f(k)(0)
k! xk.