Post on 05-Apr-2022
10. Exponential and Logarithmic Functions
10.1 The Nature of Exponential Functions
In power expressions such as ๐ฅ3 or ๐ฅ5, the exponents are constants.
A function whose independent variable appears in the role of an exponent such
as 3๐ฅ is called an exponential function.
โข Simple exponential function
๐ฆ = ๐ ๐ก = ๐๐ก (1)
๐ฆ : the dependent variables, ๐ก : the independent variable,
๐ : a fixed base of the exponent.
โข Generalized Exponential Function
๐ฆ = ๐ ๐ก = ๐๐๐๐ก (2)
๐ฆ
๐ก0
๐ฆ
๐ก0 2๐ก๐ก๐ก0
1
๐ฆ = ๐๐ก
๐ฆ = ๐2๐ก
๐ก0
๐ฆ = ๐๐ก
๐ฆ = 2๐๐ก
4
2
1
Figure 10.2
โข A Preferred Base (Napierโs constant)
๐ = 2.71828 . . .
โข Natural Exponential Function
๐ฆ = ๐๐ก , ๐ฆ = ๐3๐ก , ๐ฆ = ๐ด๐๐๐ก
๐ฆ = exp ๐ก , ๐ฆ = exp 3๐ก , ๐ฆ = ๐ดexp ๐๐ก ,
โข The Derivative of Natural Exponential Function
๐
๐๐ก๐๐ก = ๐๐ก ln ๐
๐
๐๐ก๐๐ก = ๐๐ก ,
๐
๐๐ก๐ด๐๐๐ก = ๐๐ด๐๐๐ก
10.2 Natural Exponential Functions and the Problem of Growth
โข The Number ๐
Let
๐ ๐ = 1 +1
๐
๐. (3)
The function ๐ ๐ is increasing in ๐.
๐ 1 = 1 +1
1
1= 2
๐ 2 = 1 +1
2
2= 2.25
๐ 3 = 1 +1
3
3= 2.37037โฆ
๐ 4 = 1 +1
4
4= 2.44141โฆ
โฎ
The function of ๐ ๐ is bounded from above.
๐ ๐ = 1 +๐1
1
๐+
๐2
1
๐2 + โ โ โ +๐๐
1
๐๐
= 1 +1
1!+
1
2!1 โ
1
๐+ โ โ โ +
1
๐!1 โ
1
๐โ โ โ 1 โ
๐โ1
๐
โค 1 +1
1!+
1
2!+
1
3!+ โ โ โ +
1
๐!
โค 1 + 1 +1
22+
1
23+ โ โ โ +
1
2๐โ1
= 1 +1โ
1
2๐
1โ1
2
< 1 +1
1โ1
2
= 1 + 2 = 3 (4)
โข ๐ ๐ is bounded from above (๐ ๐ < 3)
โข ๐ ๐ is monotonically increasing in ๐
โ ๐ ๐ โ a certain number as ๐ โ โ.
Definition of ๐ : ๐ โก lim๐โโ
๐ ๐ = 2.71828. . . (5)
โข The approximation value of ๐
Consider the Maclaurin series of ๐ ๐ฅ = ๐๐ฅ.
๐ ๐ฅ = ๐ 0 +๐โฒ 0
1!๐ฅ +
๐โฒโฒ 0
2!๐ฅ2 + โ โ โ +
๐ ๐ 0
๐!๐ฅ๐ + ๐ ๐
= 1 + ๐ฅ +1
2!๐ฅ2 + โ โ โ +
1
๐!๐ฅ๐ + ๐ ๐, (6)
where ๐ ๐ =๐ ๐+1 ๐
๐+1 !๐ฅ๐+1 =
๐๐
๐+1 !๐ฅ๐+1 (0 < ๐ < ๐ฅ). (7)
Since ๐ ๐ โ 0 as ๐ โ โ, we have
๐๐ฅ = 1 + ๐ฅ +1
2!๐ฅ2 +
1
3!๐ฅ3 +
1
4!๐ฅ4 +
1
5!๐ฅ5 +โ โ โ (8)
Substituting ๐ฅ = 1, we find that
๐ = 1 + 1 +1
2!+1
3!+1
4!+1
5!+โ โ โ
= 2 + 0.5 + 0.1666667 + โ โ โ
โ 2.7182819 (9)
โข An Economic Interpretation of ๐
The number ๐ can be interpreted as the result of a special process of interest
compounding.
Suppose that, starting out with a principal of $1, we find a banker to offer us
the interest rate of 100% per annum. If interest is to be compounded once a
year, the value of our asset at the end of the year will be $2.
๐ 1 = initial principal ร 1 + interest rate
= 1 ร 1 +100%
1
1
= 2(10)
Suppose that interest is compounded semiannually. Then, we have
๐ 2 = 1 +100%
2ร 1 +
100%
2= 1 +
1
2
2(11)
If the frequency of compounding in 1 year is ๐, our year end asset value is
๐ ๐ = 1 +1
๐
๐(12)
When ๐ โ โ, the value of the asset at the end of 1 year will be
lim๐โโ
๐ ๐ = ๐ (13)
The number of ๐ can be interpreted as the year-end value to which a principal of
$1 will grow if interest at the rate of 100% per annum is compounded
continuously.
10.3 Logarithms
โข The Meaning of Logarithm
The log of ๐ฆ to the base ๐ is the power to which the base ๐ must be raised to attain the value ๐ฆ.
๐ฆ = ๐๐ก โ ๐ก = log๐ ๐ฆ (14)
ln ๐ฅ โ log๐ ๐ฅ.
Examples log4 16 = log4 42 = 2 log10 1000 = log10 10
3 = 3
log10 0.01 = log10 10โ2 = โ2
ln ๐2 = log๐ ๐2 = 2 ln 1 = log๐ ๐
0 = 0
ln1
๐= log๐ ๐
โ1 = โ1
โข Rules of Logarithms
Rule I : ln ๐ข๐ฃ = ln ๐ข + ln ๐ฃ ๐ข, ๐ฃ > 0Rule II : ln ๐ข/๐ฃ = ln ๐ข โ ln ๐ฃ ๐ข, ๐ฃ > 0Rule III : ln ๐ข๐ = ๐ ln ๐ข ๐ข > 0Rule IV : log๐ ๐ข = log๐ ๐ log๐ ๐ข ๐ข > 0
Rule V : log๐ ๐ =1
log๐ ๐
Proof of Rule I.
๐ข๐ฃ = ๐ln ๐ข๐ln ๐ฃ = ๐ln ๐ข+ln ๐ฃ and ๐ข๐ฃ = ๐ln ๐ข๐ฃ
โ ln๐ข๐ฃ = ln ๐ข + ln ๐ฃ
10.4 Logarithm Functions
โข Log Functions and Exponential Functions
Log functions are inverse functions of certain exponential functions.
๐ก = log๐ ๐ฆ and ๐ก = ln ๐ฆ (15)
๐ฆ = ๐๐ก and ๐ฆ = ๐๐ก (16)
Inverse function
โข The Graphical Form
๐ฆ = ๐๐ก and ๐ก = ln ๐ฆ are drawn as follows.
๐ก
๐ฆ0
๐ฆ
๐ก0
45ยฐ
1
๐ฆ = ๐๐ก
๐ก = ln ๐ฆ
Figure 10.3 mirror-relationship
45ยฐ
1
We consider the inverse of ๐ฆ = ๐ด๐๐๐ก.
Taking the natural log of both sides of this exponential function,
ln ๐ฆ = ln ๐ด๐๐๐ก = ln๐ด + ๐๐ก ln ๐ = ln๐ด + ๐๐ก, (17)
Solving for ๐ก,
๐ก =ln ๐ฆโln ๐ด
๐.
(18)
As inverse function of monotonically increasing functions, logarithmic
functions must also be monotonically increasing.
ln ๐ฆ1 = ln ๐ฆ2 โบ ๐ฆ1 = ๐ฆ2
ln ๐ฆ1 > ln ๐ฆ2 โบ ๐ฆ1 > ๐ฆ2 (19)
For any base ๐ > 1,
แ
0 < ๐ฆ < 1๐ฆ = 1๐ฆ > 1
โบ เต
log๐ ๐ฆ < 0log๐ ๐ฆ = 0log๐ ๐ฆ > 0
(20)
โข Base Conversion
Let us consider the conversion of ๐ด๐๐๐ก into ๐ด๐๐๐ก.
๐๐ = ๐๐ โน ln๐๐ = ln ๐๐
โน ๐ = ๐ ln ๐ (21)
Thus,
๐ด๐๐๐ก = ๐ด๐ ๐ ln ๐ ๐ก
10.5 Derivative of Exponential and Logarithmic Functions
โข Log-Function Rule
๐
๐๐กln ๐ก =
1
๐ก(23)
โข Exponential-Function Rule
๐
๐๐ก๐๐ก = ๐๐ก (24)
โข The Rules Generalized
๐
๐๐ก๐๐ ๐ก = ๐โฒ ๐ก ๐๐ ๐ก
๐
๐๐กln ๐ ๐ก =
๐โฒ ๐ก
๐ ๐ก(25)