Exponential functions in Economics:
Economics applications of ln
Interest compounding: for an interest rate r compounded Alternative definition of rate of growth: since the rate of
at frequency m on an initial principal
A,the value of the growth is 1 df = d ln (f (t)) , it can also be expressed as
r mt
f (t) dt
dt
. In the limit
asset at time t is V (m, t) = A 1 +
d ln (f (t))
m
.
m → ∞ we have: V (t) = Aert . The rate r can take
dt
negative values in the case of deflation or depreciation.
Elasticity : the elasticity of a function y(x) with respect to
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d ln y
x dy
Rate of growth: for a function f (t), the rate of growth is
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x is
=
. If y is an exponential function of x,
1 df
d
ln
x
y dx
. In the case f (t) represents an expodefined as
then the elasticity is the slope of the straight line obtained
f (t) dt
mcccp-richard-4
nential growth and takes the form Aert then the rate of when plotting y as a function of x on a log-log graph (which
is the same as plotting ln y as a function of ln x).
Arert
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growth is
= r.
Aert
Exponential and Logarithm for Economics and
The Cobb-Douglas Production functions: are widely com5
Business Studies
mon in Economics and are a family of functions taking the
exp(x)
4
This leaflet is an overview of the properties of the functions form: Q = AK α Lβ .
3
e and ln and their applications in Economics.
Logarithm
2
Author: Morgiane Richard, University of Aberdeen
1
The logarithm function log in base b is the inverse function
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mathcentre community project
ln(x)
Reviewers: Anthony Cronin (University College Dublin), of the exponential function in base b:
0
y
Shazia Ahmed (University of Glasgow)
−1
y = logb x ⇔ x = b
−2
Exponential
The natural logarithm, ln is the inverse function of the
−3
The exponential function is f (t) = bt , where b > 1 is called natural exponential function:
−4
the base.
y = ln x ⇔ x = ey
−5
−1
0
1
2
3
4
5
The most commonly occurring base in Business and Ecoln
x
x
nomics is e ≈ 2.72 and the corresponding exponential func- This means that e
= ln e = x and for any base b,
tion is the natural exponential functionf (t) = et = exp (t). blogb x = logb bx = x.
Figure 1: Graph of the functions ex and ln x.
m
1
Properties of ln:
The number e is defined as e = lim 1 +
.
m→∞
m
ln (ut) = ln u + ln t
Properties of e:
1
t
ln ( ) = − ln t and ln ( ) = ln t − ln u
t u
u t
ut
www.mathcentre.ac.uk
t
u
(e ) = (e ) = e
u
ln
(t
)
=
u
ln
t
t+u
t u
e
=ee
e−t =
1
et
t−u
and
e
=
et
eu
d(et )
d(ef (t) )
= et and
= f 0 (t)ef (t)
dt
dt
logb t = logc t logb c and logb t = ln t logb e (conversion of
base)
d ln(t)
1
d ln (f (t))
f 0 (t)
1 df (t)
= and
=
=
dt
t
dt
f (t)
f (t) dt
1