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ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
BEE1024 Mathematics for EconomistsExponential and logarithmic functions, Elasticities
Juliette Stephenson and Amr (Miro) AlgarhiAuthor: Dieter Balkenborg
Department of Economics, University of Exeter
Week 5
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
1 Objectives
2 The Exponential FunctionDe�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Objectives
Exponential functions: describes growth processes withconstant growth ratepopulation growth, growth of GDP, in�ation etc...
logarithm: the exponent required to produce a given numberinverse function, transforms multiplication into addition:10a � 10b = 10a+bLogarithmic di¤erentiation
Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Objectives
Exponential functions: describes growth processes withconstant growth ratepopulation growth, growth of GDP, in�ation etc...
logarithm: the exponent required to produce a given numberinverse function, transforms multiplication into addition:10a � 10b = 10a+bLogarithmic di¤erentiation
Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Objectives
Exponential functions: describes growth processes withconstant growth ratepopulation growth, growth of GDP, in�ation etc...
logarithm: the exponent required to produce a given numberinverse function, transforms multiplication into addition:10a � 10b = 10a+bLogarithmic di¤erentiation
Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
The Exponential Function
power: xy
base: x
index or exponent: y
power function like x2: vary x
exponential function 2y : vary y
admissible values for y : positive integers, integers, rationals,real numbers
problem: for general y the power xy can only be de�ned forpositive x
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
The Exponential Function
power: xy
base: x
index or exponent: y
power function like x2: vary x
exponential function 2y : vary y
admissible values for y : positive integers, integers, rationals,real numbers
problem: for general y the power xy can only be de�ned forpositive x
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
The Exponential Function
power: xy
base: x
index or exponent: y
power function like x2: vary x
exponential function 2y : vary y
admissible values for y : positive integers, integers, rationals,real numbers
problem: for general y the power xy can only be de�ned forpositive x
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
The Exponential Function
power: xy
base: x
index or exponent: y
power function like x2: vary x
exponential function 2y : vary y
admissible values for y : positive integers, integers, rationals,real numbers
problem: for general y the power xy can only be de�ned forpositive x
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
The Exponential Function
power: xy
base: x
index or exponent: y
power function like x2: vary x
exponential function 2y : vary y
admissible values for y : positive integers, integers, rationals,real numbers
problem: for general y the power xy can only be de�ned forpositive x
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
The Exponential Function
power: xy
base: x
index or exponent: y
power function like x2: vary x
exponential function 2y : vary y
admissible values for y : positive integers, integers, rationals,real numbers
problem: for general y the power xy can only be de�ned forpositive x
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
The Exponential Function
power: xy
base: x
index or exponent: y
power function like x2: vary x
exponential function 2y : vary y
admissible values for y : positive integers, integers, rationals,real numbers
problem: for general y the power xy can only be de�ned forpositive x
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Power functions
0
2468
1012141618202224
4 2 2 4x
y = 2: f (x) = x2
0
1
2
3
4
4 2 2 4x
y = �2:f (x) = x�2 = 1
x 2
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Power functions
0
0.20.40.60.8
11.21.41.61.8
22.2
1 2 3 4 5x
y = 12 : f (x) = x
12 =
px
0
1
2
3
4
5
1 2 3 4 5x
y = � 32 :
f (x) = x�32 = 1
xpx
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Exponential Functions
approximate irrational index y by fraction mn :
xy := limmn!y
xmn .
0
2
4
6
8
2 1 1 2y
x = 3: g (y) = 3y
0
2
4
6
8
2 1 1 2y
x = 13 : g (y) =
� 13
�y= 3�y
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Exponential Functions
0.51.5 2 2.5 3
x0.6
0.81y
1
2
3
4
z = xy x � 0Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Properties of exponential functions
An exponential function ax :
is strictly convex and has strictly positive values;
is for a > 1 strictly increasing with limx!�∞ ax = 0 andlimx!∞ ax = +∞;is for 0 < a < 1: decreasing with limx!�∞ ax = +∞ andlimx!∞ ax = 0.
Calculational rules for generalized powers:
as+t = asat ast = (as )t (ab)s = asbs
but(as )t 6= a(s t )
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Properties of exponential functions
An exponential function ax :is strictly convex and has strictly positive values;
is for a > 1 strictly increasing with limx!�∞ ax = 0 andlimx!∞ ax = +∞;is for 0 < a < 1: decreasing with limx!�∞ ax = +∞ andlimx!∞ ax = 0.
Calculational rules for generalized powers:
as+t = asat ast = (as )t (ab)s = asbs
but(as )t 6= a(s t )
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Properties of exponential functions
An exponential function ax :is strictly convex and has strictly positive values;
is for a > 1 strictly increasing with limx!�∞ ax = 0 andlimx!∞ ax = +∞;
is for 0 < a < 1: decreasing with limx!�∞ ax = +∞ andlimx!∞ ax = 0.
Calculational rules for generalized powers:
as+t = asat ast = (as )t (ab)s = asbs
but(as )t 6= a(s t )
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Properties of exponential functions
An exponential function ax :is strictly convex and has strictly positive values;
is for a > 1 strictly increasing with limx!�∞ ax = 0 andlimx!∞ ax = +∞;is for 0 < a < 1: decreasing with limx!�∞ ax = +∞ andlimx!∞ ax = 0.
Calculational rules for generalized powers:
as+t = asat ast = (as )t (ab)s = asbs
but(as )t 6= a(s t )
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Compounded interests and the number e
Put P0 > 0 (the principal) in savings account
�xed nominal annual interests rate r > 0
Interests paid n times during the year
amount Pt in your savings account after t years:
formula for compounded interests
Pt = P0�1+ r
n
�ntrn interest paid per period
nt total number of interest payments.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Compounded interests and the number e
Put P0 > 0 (the principal) in savings account
�xed nominal annual interests rate r > 0
Interests paid n times during the year
amount Pt in your savings account after t years:
formula for compounded interests
Pt = P0�1+ r
n
�ntrn interest paid per period
nt total number of interest payments.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Compounded interests and the number e
Put P0 > 0 (the principal) in savings account
�xed nominal annual interests rate r > 0
Interests paid n times during the year
amount Pt in your savings account after t years:
formula for compounded interests
Pt = P0�1+ r
n
�ntrn interest paid per period
nt total number of interest payments.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Compounded interests and the number e
Put P0 > 0 (the principal) in savings account
�xed nominal annual interests rate r > 0
Interests paid n times during the year
amount Pt in your savings account after t years:
formula for compounded interests
Pt = P0�1+ r
n
�ntrn interest paid per period
nt total number of interest payments.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Compounded interests and the number e
Put P0 > 0 (the principal) in savings account
�xed nominal annual interests rate r > 0
Interests paid n times during the year
amount Pt in your savings account after t years:
formula for compounded interests
Pt = P0�1+ r
n
�nt
rn interest paid per period
nt total number of interest payments.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Compounded interests and the number e
Put P0 > 0 (the principal) in savings account
�xed nominal annual interests rate r > 0
Interests paid n times during the year
amount Pt in your savings account after t years:
formula for compounded interests
Pt = P0�1+ r
n
�ntrn interest paid per period
nt total number of interest payments.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Compounded interests and the number e
Put P0 > 0 (the principal) in savings account
�xed nominal annual interests rate r > 0
Interests paid n times during the year
amount Pt in your savings account after t years:
formula for compounded interests
Pt = P0�1+ r
n
�ntrn interest paid per period
nt total number of interest payments.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Compounded interests and the number e
The (natural) exponential function:balance in account after one year if interests paid continuously:
exp (r) = limn!∞�1+ r
n
�n.
The table below shows the value of�1+ r
n
�n for various n and r :r = 5.4% r = 5.5% r = 100%
n = 4 1.055103375 1.056144809 2.44140625n = 12 1.055356752 1.05640786 2.61303529n = 364 1.055480375 1.056536225 2.714557303n = 8736 1.055484426 1.056540432 2.718126265n = 524 160 1.055484599 1.056540612 2.718279235n = 31 449 600 1.055484602 1.056540613 2.718281796n! +∞ 1.055484602 1.056540615 2.718281828
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Compounded interests and the number e
The Euler number e is de�ned as
e = exp (1) = limn!∞�1+ 1
n
�n.
The �natural exponential function�is indeed the exponentialfunction with base e:
exp (r) = er
�proof� for rational r :
exp (r) = limn!∞
�1+
rn
�n= lim
m!∞,n=rm
�1+
rn
�n= lim
m!∞
�1+
1m
�rm= lim
m!∞
��1+
1m
�m�r=
�limm!∞
�1+
1m
�m�r= er
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Compounded interests and the number e
formula for continuously compounded interests:
Pt = P0ert .
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Properties of the natural exponential function
1 e0 = 1,2 e1 = e3 ex > 0 for all x4
d (ex )dx = ex
In particular, ex is strictly increasing and convex.instantaneous growth rate of a function y = f (x): dy
dx
.y
when x is increased by a exponential function has constantgrowth rate 1.
5
ea+b = eaeb (ea)b = eab .
In particular 1ex = e
�x (because e�x ex = e0 = 1).
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Let fn (x) =�1+ x
n
�n.Intuition for property 3:
�1+ x
n
�is positive when x > 0 or when n
large compared to jx j . Then fn (x) > 0 and so hence ex > 0.Intuition for property 4:
dfndx
= n�1+
xn
�n�1 1n=�1+
xn
�n�1��1+
xn
�n= fn (x)
for n very large compared to jx j since 1+ xn is then very close to 1.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Theorem
There is one and only one function y = f (x) which satis�es the�initial condition� f (0) = 1 and the �di¤erential equation�
dydx= y
and this is the exponential function f (x) = exp x = ex .
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Properties of the natural exponential function
Exponential versus polynomial growth: For any polynomialP (x)
limx!+∞
ex
P (x)= +∞
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
Intuition: Suppose P (x) = amxm + . . . has degree m.Approximate ex by
�1+ x
n
�n with n larger than m. Thenlim
x!+∞
ex
P (x)� lim
x!+∞
�1+ x
n
�nP (x)
= limx!+∞
� 1n
�nxn + . . .
amxm + . . .=
limx!+∞
Cxn�m = +∞
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
A quicker way to calculate ex : is to use the formula
ex = 1+ x +x2
2!+x3
3!+ . . .+
xn
n!+ . . .
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Logarithmic functions
Let a > 0. The logarithmic function loga x to the base a is de�nedas the inverse of the exponential function ay
y = loga x , ay = x
For instance, 1000 = 103, so log10 1000 = 3;18 = 2
�3, solog2
� 18
�= �3.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
The natural logarithm
natural logarithm function
y = ln (x), x = ey
3
2
1
0
2
3
4
3 2 1 1 2 3 4x
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Di¤erentiating the natural logarithm
x = e ln x
(ln x)0 =1e ln x
=1x= x�1
because by the chain rule
1 =dxdx= e ln x � (ln x)0 = x � (ln x)0
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Properties of the natural logarithm
1 ln (y) is only de�ned for strictly positive y > 0.
2d ln(y )dy = 1
y . In particular, ln (y) is strictly increasing andconcave.
3 ln (1) = 0, ln (e) = 1.4 limy!0 ln (y) = �∞, limy!+∞ ln (y) = +∞.5 ln (ab) = ln (a) + ln (b), ln
�ab�= b ln (a). In particular
ln� 1a
�= � ln (a).
6 Logarithmic differentiation: Combined with the chainrule one obtains the following useful formula where y = g (x)is any di¤erentiable function:
d ln(g (x ))dx = g 0(x )
g (x ) (1)
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Properties of the natural logarithm
1 ln (y) is only de�ned for strictly positive y > 0.2
d ln(y )dy = 1
y . In particular, ln (y) is strictly increasing andconcave.
3 ln (1) = 0, ln (e) = 1.4 limy!0 ln (y) = �∞, limy!+∞ ln (y) = +∞.5 ln (ab) = ln (a) + ln (b), ln
�ab�= b ln (a). In particular
ln� 1a
�= � ln (a).
6 Logarithmic differentiation: Combined with the chainrule one obtains the following useful formula where y = g (x)is any di¤erentiable function:
d ln(g (x ))dx = g 0(x )
g (x ) (1)
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Properties of the natural logarithm
1 ln (y) is only de�ned for strictly positive y > 0.2
d ln(y )dy = 1
y . In particular, ln (y) is strictly increasing andconcave.
3 ln (1) = 0, ln (e) = 1.
4 limy!0 ln (y) = �∞, limy!+∞ ln (y) = +∞.5 ln (ab) = ln (a) + ln (b), ln
�ab�= b ln (a). In particular
ln� 1a
�= � ln (a).
6 Logarithmic differentiation: Combined with the chainrule one obtains the following useful formula where y = g (x)is any di¤erentiable function:
d ln(g (x ))dx = g 0(x )
g (x ) (1)
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Properties of the natural logarithm
1 ln (y) is only de�ned for strictly positive y > 0.2
d ln(y )dy = 1
y . In particular, ln (y) is strictly increasing andconcave.
3 ln (1) = 0, ln (e) = 1.4 limy!0 ln (y) = �∞, limy!+∞ ln (y) = +∞.
5 ln (ab) = ln (a) + ln (b), ln�ab�= b ln (a). In particular
ln� 1a
�= � ln (a).
6 Logarithmic differentiation: Combined with the chainrule one obtains the following useful formula where y = g (x)is any di¤erentiable function:
d ln(g (x ))dx = g 0(x )
g (x ) (1)
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Properties of the natural logarithm
1 ln (y) is only de�ned for strictly positive y > 0.2
d ln(y )dy = 1
y . In particular, ln (y) is strictly increasing andconcave.
3 ln (1) = 0, ln (e) = 1.4 limy!0 ln (y) = �∞, limy!+∞ ln (y) = +∞.5 ln (ab) = ln (a) + ln (b), ln
�ab�= b ln (a). In particular
ln� 1a
�= � ln (a).
6 Logarithmic differentiation: Combined with the chainrule one obtains the following useful formula where y = g (x)is any di¤erentiable function:
d ln(g (x ))dx = g 0(x )
g (x ) (1)
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Properties of the natural logarithm
1 ln (y) is only de�ned for strictly positive y > 0.2
d ln(y )dy = 1
y . In particular, ln (y) is strictly increasing andconcave.
3 ln (1) = 0, ln (e) = 1.4 limy!0 ln (y) = �∞, limy!+∞ ln (y) = +∞.5 ln (ab) = ln (a) + ln (b), ln
�ab�= b ln (a). In particular
ln� 1a
�= � ln (a).
6 Logarithmic differentiation: Combined with the chainrule one obtains the following useful formula where y = g (x)is any di¤erentiable function:
d ln(g (x ))dx = g 0(x )
g (x ) (1)
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Theorem
The natural logarithm y (x) = ln x is the unique solution to thedi¤erential equation
dydx=1x
which satis�es the initial condition y (1) = 0.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Di¤erentiating general exponential and logarithmicfunctions
ln (xy ) = y ln (x)general powers can be rewritten as
xy = ey ln(x ) = e index�ln(base).
Partial di¤erentiation yields
∂xy
∂y= ey ln(x ) ln (x) = xy ln (x)
∂xy
∂x= ey ln(x )
�y1x
�= yxy
1x= yxy�1.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
Derivative of an exponential function y = ax is
daxdx = ln (a) a
x .
The instantaneous growth rate of y = ax is ln (a)The derivative of a power function y = xb is
dx bdx = bx
b�1
even if b is irrational.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
De�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
x = loga (y), y = ax .
We have
y = ax , y = ex ln(a) , ln (y) = x ln (a), x =ln (y)ln (a)
so
loga y =ln(y )ln(a)
loga y has hence the derivative
d (loga(y ))dy = 1
ln(a)y
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
Compounded Interests
Example: Suppose Bank A o¤ers the annual nominal interest raterA = 5.5% and pays interests monthly. Bank B o¤ers the annualnominal interest rate rB = 5.4% and pays interests daily. Whichbank o¤ers the better deal?Solution:
reff ,A =�1+
rA12
�12� 1 =
�1+
0.05512
�12� 1 = 5.64%
reff ,B =�1+
rB364
�364� 1 =
�1+
0.054364
�364� 1 = 5.55%
so Bank A o¤ers better deal.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
Exponential decay
most radioactive substances decay exponentiallysample of initial size Q0 weights Q (t) = Q0e�kt at time t.k measures rate of decay, �half-life� of the radioactive substance:Example: Show that a radioactive substance that decaysaccording to the formula Q (t) = Q0e�kt has a half-life of t̄ = ln 2
k .Solution: �nd value t̄ for which Q (t̄) = 1
2Q0, that is
12Q0 = Q0e�k t̄ .
Divide by Q0 and take natural logarithm:
ln12= �kt̄.
Thus the half-life is
t̄ =� ln 12k
=ln 2k
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
Di¤erentiation
Example: Find the derivative of g (x) = xx .Solution: Using logarithmic di¤erentiation we obtain
g 0 (x)g (x)
=d (ln xx )dx
=d�ln ex ln(x )
�dx
=d (x ln (x))
dx
= 1� ln (x) + x � 1x= ln (x) + 1
g 0 (x) = (ln (x) + 1) xx .
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
The logistic curve
The graph of the function of the form
Q (t) =B
1+ Ae�Bkt
where A, B, k are positive constants, is called a logistic curve.describes growth processes when environmental factors impose a�braking� e¤ect on the rate of growth.Example: Show that the growth rate of the logistic curveQ (t) = 1
1+e�t is 1�Q (t).
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
Solution: We have
Q 0 (t) =�e�t (�1)(1+ e�t )2
=e�t
(1+ e�t )2
Q 0 (t)Q (t)
=e�t
1+ e�t
1�Q (t) =1+ e�t � 11+ e�t
=e�t
1+ e�t
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
Theorem
The logistic curve y (x) = 11+e�x is the unique solution to the
di¤erential equationdydx= y (1� y)
which satis�es the initial condition y (0) = 12 .
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
Example
Public health records indicate that t weeks after the outbreak of acertain form of in�uenza, approximately Q (t) = 20
1+19e�1.2t
thousand people had caught the disease.a) How many people had the disease when it broke out? Howmany had it two weeks later?b) At what time does the spread of the infection begin to decline?c) If the trend continues, approximately how many people willeventually contract the disease?
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
Solution: a) Since Q (0) = 201+19 = 1 it follows that 1000 people
initially had the disease. When t = 2
Q (2) =20
1+ 19e�2.4� 7.343
so about 7.343 thousand people had contracted the disease by thesecond week.b) in�ection point at t̄ � 2.5. For t < t̄ convex and so the numberof newly infected increasing. For t > t̄ concave and so the numberof newly infected is decreasing.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Compounded InterestsExponential decayThe logistic curve
0
5
10
15
4 2 2 4 6 8t
c) limt!+∞ Q (t) = 20, so roughly 20000 people catch the diseaseon total.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
ElasticitiesOwn-price elasticity
demand function
Qd = 1000� P3dQd
dP= �3P2
current price is £ 5, price raised by a pound:quantity demand decreases approximately bydQ ddP jP=5 = 3� 5
2 = 75 tons.
one-percent increase in the price: increase by 5p= 120 � $1
reduce quantity demanded by approximately 3.75 = 7520 tons.
demand at £ 5 is 1000� 53 = 875percentage decrease in quantity demanded is 3.75875 �0.0043 = 0.43%.Thus 1% increase in price reduces quantity demanded by 0.43%.Demand is inelastic at this price.Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
More generally: dQd
dP approximate change in demand when theprice increases by a pound.initial price is P,increase by one pound is an increase by 100
P percent.an increase of the price by 1% changes the quantity demandedby approximately by P
100 �dQ ddP tons.
percentage change in quantity demanded is approximately:
ped (P) =100Qd
� P100
� dQd
dP=dQd
dP� PQd
This is the own-price elasticity of demand.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
rewrite this formula as
ped (P) =dQd
Qd� dPP
where 100dPP is the percentage increase in price and 100dQd
Q d is(approximately) the induced percentage change in quantity.In our example
ped (P) =��3P2
�� PQd
= �3 P3
1000� P3
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
When is demand inelastic?
3P3
1000� P3 < 1
or
3P3 < 1000�P3 4P3 < 1000 P3 < 250 P < 3p250 � 6.3
Exactly when P = 3p250 there is unit elasticity and above demand
is elastic.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
Constant Elasticity
A function has constant elasticity ε if and only if it is of the form
Q = αP ε
We have
dQdP
= αεP ε�1
dQdP
PQ
= αεP ε�1 PαP ε
= ε
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
Total Revenue
With this demand, total revenue of the market is
TR = PQd = P�1000� P3
�Total revenue is maximized when
dTRdP
= 1000� 4P3 = 0
or P = 3p250, i.e., exactly when there is unit elasticity.
Since d 2TRdP 2 = �12P
2 < 0, total revenue decreases to the left andincreases to the right of this price.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
Inverse Demand
The consumers will demand a quantity Q when the price P is suchthat
Q = Qd (P) = 1000� P3
P3 = 1000�QPd = 3
p1000�Q
inverse demand function
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
Marginal Revenue
use inverse demand function to express total revenue as a functionof quantity:
TR = PQ = 3p1000�Q �Q = (1000�Q)
13 �Q
quantity demanded is decreasing in price.total revenue is increasing in price when it decreasing in quantityand vice versa.
Marginal revenue is the change in revenue if a small unit more ofthe commodity is sold on the market.
MR =dTRdQ
= �13(1000�Q)�
23 �Q + (1000�Q)
13
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
Marginal revenue is zero when
(1000�Q)�23 �Q = 3 (1000�Q)
13
Q = 3 (1000�Q) = 3000� 3Q4Q = 3000 Q = 750
at P = 3p250. For lower quantities it is positive and for higher
ones negative.
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
TheoremMarginal revenue and own-price elasticity are related by
MR = P�1+
1ped (P)
�
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
This is so because
dTRdP
=d�PQd
�dP
= Qd + PdQd
dP
whereas by the chain rule
dTRdP
=dTRdQ
dQd
dP
and so
MR =dTRdQ
=
�Qd + P
dQd
dP
��dQd
dP=Qd
dQ ddP
+P =P
dQ ddP
PQ d
+P
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
Elasticities and logarithms
data (x , y) = (lnP, lnQ). Then
dydx= ped (P)
dydQ =
1Q , P = e
x dPdx = e
x = P.The chain rule applied twice yields
dydx=dydQ
dQdP
dPdx=1QdQdPP = ped (P)
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
Gliederung1 Objectives2 The Exponential Function
De�nitionProperties of exponential functionsCompounded interests and the number eProperties of the natural exponential function
3 Logarithmic functionsDe�nitionDi¤erentiating the natural logarithmProperties of the natural logarithmDi¤erentiating general exponential and logarithmic functions
4 ApplicationsCompounded InterestsExponential decayThe logistic curve
5 ElasticitiesOwn-price elasticityElasticities and logarithmsOther Elasticities
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
Other Elasticities
demand for a commodity function of own price, income and otherprices.
Qd = 100� p + 2p� � 3yp is the price of the commodity,p� the price of another commodityy is income.
own price elasticity:∂Qd
∂ppQd
= � pQd
Balkenborg Exponential and logarithmic functions, Elasticities
ObjectivesThe Exponential Function
Logarithmic functionsApplicationsElasticities
Own-price elasticityElasticities and logarithmsOther Elasticities
Other Elasticities
demand for a commodity function of own price, income and otherprices.
Qd = 100� p + 2p� � 3ycross price elasticity:
∂Qd
∂p�p�
Qd= 2
p�
Qd
income elasticity∂Qd
∂yyQd
= �3 yQd
Balkenborg Exponential and logarithmic functions, Elasticities