10. Exponential and Logarithmic Functions

10.1 The Nature of Exponential Functions

In power expressions such as 𝑥3 or 𝑥5, the exponents are constants.

A function whose independent variable appears in the role of an exponent such

as 3𝑥 is called an exponential function.

• Simple exponential function

𝑦 = 𝑓 𝑡 = 𝑏𝑡 (1)

𝑦 : the dependent variables, 𝑡 : the independent variable,

𝑏 : a fixed base of the exponent.

• Generalized Exponential Function

𝑦 = 𝑓 𝑡 = 𝑎𝑏𝑐𝑡 (2)

𝑦

𝑡0

𝑦

𝑡0 2𝑡𝑡𝑡0

1

𝑦 = 𝑏𝑡

𝑦 = 𝑏2𝑡

𝑡0

𝑦 = 𝑏𝑡

𝑦 = 2𝑏𝑡

4

2

1

Figure 10.2

• A Preferred Base (Napier’s constant)

𝑒 = 2.71828 . . .

• Natural Exponential Function

𝑦 = 𝑒𝑡 , 𝑦 = 𝑒3𝑡 , 𝑦 = 𝐴𝑒𝑟𝑡

𝑦 = exp 𝑡 , 𝑦 = exp 3𝑡 , 𝑦 = 𝐴exp 𝑟𝑡 ,

• The Derivative of Natural Exponential Function

𝑑

𝑑𝑡𝑏𝑡 = 𝑏𝑡 ln 𝑏

𝑑

𝑑𝑡𝑒𝑡 = 𝑒𝑡 ,

𝑑

𝑑𝑡𝐴𝑒𝑟𝑡 = 𝑟𝐴𝑒𝑟𝑡

10.2 Natural Exponential Functions and the Problem of Growth

• The Number 𝑒

Let

𝑓 𝑚 = 1 +1

𝑚

𝑚. (3)

The function 𝑓 𝑚 is increasing in 𝑚.

𝑓 1 = 1 +1

1

1= 2

𝑓 2 = 1 +1

2

2= 2.25

𝑓 3 = 1 +1

3

3= 2.37037…

𝑓 4 = 1 +1

4

4= 2.44141…

⋮

The function of 𝑓 𝑚 is bounded from above.

𝑓 𝑚 = 1 +𝑚1

1

𝑚+

𝑚2

1

𝑚2 + ∙ ∙ ∙ +𝑚𝑚

1

𝑚𝑚

= 1 +1

1!+

1

2!1 −

1

𝑚+ ∙ ∙ ∙ +

1

𝑚!1 −

1

𝑚∙ ∙ ∙ 1 −

𝑚−1

𝑚

≤ 1 +1

1!+

1

2!+

1

3!+ ∙ ∙ ∙ +

1

𝑚!

≤ 1 + 1 +1

22+

1

23+ ∙ ∙ ∙ +

1

2𝑚−1

= 1 +1−

1

2𝑚

1−1

2

< 1 +1

1−1

2

= 1 + 2 = 3 (4)

• 𝑓 𝑚 is bounded from above (𝑓 𝑚 < 3)

• 𝑓 𝑚 is monotonically increasing in 𝑚

⇒ 𝑓 𝑚 → a certain number as 𝑚 → ∞.

Definition of 𝒆 : 𝑒 ≡ lim𝑚→∞

𝑓 𝑚 = 2.71828. . . (5)

• The approximation value of 𝑒

Consider the Maclaurin series of 𝜙 𝑥 = 𝑒𝑥.

𝜙 𝑥 = 𝜙 0 +𝜙′ 0

1!𝑥 +

𝜙′′ 0

2!𝑥2 + ∙ ∙ ∙ +

𝜙 𝑛 0

𝑛!𝑥𝑛 + 𝑅𝑛

= 1 + 𝑥 +1

2!𝑥2 + ∙ ∙ ∙ +

1

𝑛!𝑥𝑛 + 𝑅𝑛, (6)

where 𝑅𝑛 =𝜙 𝑛+1 𝑝

𝑛+1 !𝑥𝑛+1 =

𝑒𝑝

𝑛+1 !𝑥𝑛+1 (0 < 𝑝 < 𝑥). (7)

Since 𝑅𝑛 → 0 as 𝑛 → ∞, we have

𝑒𝑥 = 1 + 𝑥 +1

2!𝑥2 +

1

3!𝑥3 +

1

4!𝑥4 +

1

5!𝑥5 +∙ ∙ ∙ (8)

Substituting 𝑥 = 1, we find that

𝑒 = 1 + 1 +1

2!+1

3!+1

4!+1

5!+∙ ∙ ∙

= 2 + 0.5 + 0.1666667 + ∙ ∙ ∙

≅ 2.7182819 (9)

• An Economic Interpretation of 𝑒

The number 𝑒 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.

𝑉 1 = initial principal × 1 + interest rate

= 1 × 1 +100%

1

1

= 2(10)

Suppose that interest is compounded semiannually. Then, we have

𝑉 2 = 1 +100%

2× 1 +

100%

2= 1 +

1

2

2(11)

If the frequency of compounding in 1 year is 𝑚, our year end asset value is

𝑉 𝑚 = 1 +1

𝑚

𝑚(12)

When 𝑚 → ∞, the value of the asset at the end of 1 year will be

lim𝑚→∞

𝑉 𝑚 = 𝑒 (13)

The number of 𝑒 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

• The Meaning of Logarithm

The log of 𝑦 to the base 𝑏 is the power to which the base 𝑏 must be raised to attain the value 𝑦.

𝑦 = 𝑏𝑡 ⇔ 𝑡 = log𝑏 𝑦 (14)

ln 𝑥 ⇔ log𝑒 𝑥.

Examples log4 16 = log4 42 = 2 log10 1000 = log10 10

3 = 3

log10 0.01 = log10 10−2 = −2

ln 𝑒2 = log𝑒 𝑒2 = 2 ln 1 = log𝑒 𝑒

0 = 0

ln1

𝑒= log𝑒 𝑒

−1 = −1

• Rules of Logarithms

Rule I : ln 𝑢𝑣 = ln 𝑢 + ln 𝑣 𝑢, 𝑣 > 0Rule II : ln 𝑢/𝑣 = ln 𝑢 − ln 𝑣 𝑢, 𝑣 > 0Rule III : ln 𝑢𝑎 = 𝑎 ln 𝑢 𝑢 > 0Rule IV : log𝑏 𝑢 = log𝑏 𝑒 log𝑒 𝑢 𝑢 > 0

Rule V : log𝑏 𝑒 =1

log𝑒 𝑏

Proof of Rule I.

𝑢𝑣 = 𝑒ln 𝑢𝑒ln 𝑣 = 𝑒ln 𝑢+ln 𝑣 and 𝑢𝑣 = 𝑒ln 𝑢𝑣

⇒ ln𝑢𝑣 = ln 𝑢 + ln 𝑣

10.4 Logarithm Functions

• Log Functions and Exponential Functions

Log functions are inverse functions of certain exponential functions.

𝑡 = log𝑏 𝑦 and 𝑡 = ln 𝑦 (15)

𝑦 = 𝑏𝑡 and 𝑦 = 𝑒𝑡 (16)

Inverse function

• The Graphical Form

𝑦 = 𝑒𝑡 and 𝑡 = ln 𝑦 are drawn as follows.

𝑡

𝑦0

𝑦

𝑡0

45°

1

𝑦 = 𝑒𝑡

𝑡 = ln 𝑦

Figure 10.3 mirror-relationship

45°

1

We consider the inverse of 𝑦 = 𝐴𝑒𝑟𝑡.

Taking the natural log of both sides of this exponential function,

ln 𝑦 = ln 𝐴𝑒𝑟𝑡 = ln𝐴 + 𝑟𝑡 ln 𝑒 = ln𝐴 + 𝑟𝑡, (17)

Solving for 𝑡,

𝑡 =ln 𝑦−ln 𝐴

𝑟.

(18)

As inverse function of monotonically increasing functions, logarithmic

functions must also be monotonically increasing.

ln 𝑦1 = ln 𝑦2 ⟺ 𝑦1 = 𝑦2

ln 𝑦1 > ln 𝑦2 ⟺ 𝑦1 > 𝑦2 (19)

For any base 𝑏 > 1,

ቑ

0 < 𝑦 < 1𝑦 = 1𝑦 > 1

⟺ ൞

log𝑏 𝑦 < 0log𝑏 𝑦 = 0log𝑏 𝑦 > 0

(20)

• Base Conversion

Let us consider the conversion of 𝐴𝑏𝑐𝑡 into 𝐴𝑒𝑟𝑡.

𝑒𝑟 = 𝑏𝑐 ⟹ ln𝑒𝑟 = ln 𝑏𝑐

⟹ 𝑟 = 𝑐 ln 𝑏 (21)

Thus,

𝐴𝑏𝑐𝑡 = 𝐴𝑒 𝑐 ln 𝑏 𝑡

10.5 Derivative of Exponential and Logarithmic Functions

• Log-Function Rule

𝑑

𝑑𝑡ln 𝑡 =

1

𝑡(23)

• Exponential-Function Rule

𝑑

𝑑𝑡𝑒𝑡 = 𝑒𝑡 (24)

• The Rules Generalized

𝑑

𝑑𝑡𝑒𝑓 𝑡 = 𝑓′ 𝑡 𝑒𝑓 𝑡

𝑑

𝑑𝑡ln 𝑓 𝑡 =

𝑓′ 𝑡

𝑓 𝑡(25)