Chapter 10: Exponential and Logarithmic Functions

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Chapter 10Exponential and Logarithmic Functions

• Alpha Chiang, Fundamental Methods of Mathematical
  Economics
• 3rd edition

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Exponential functions
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Exponential functions
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Properties of exponential functions
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The number e
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The number e
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Economic interpretation of e

• it can be interpreted as the result of a special
  process of interest compounding.

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Economic interpretation of e

• For the limiting case, when interest is
  compounded continuously during the year, the
  value of the asset will grow in a snowballing
  fashion becoming

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Interest Compounding and the function Aert
A reflects change in principal from
previous level of P1 r/m means that in each
of the compounding periods in a year, only 1/m
of the nominal interest will actually be
applicable. mt since interest is to be
compounded m times a year, there should be a
total of mt compounding in t years.
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Interest Compounding and the function Aert
Alterative form
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Instantaneous Rate of Growth
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Discounting and Negative Growth
Discrete
Continuous
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Logarithms
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Common log
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Natural log
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Rules
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Logarithmic Functions
Logarithmic Functions are functions whose
variables are expressed as a function of the
logarithm of another variable.
Log functions are inverse functions of certain
exponential functions
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Derivatives of Exponential and Logarithmic
Functions
Log function rule
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Exponential function rule
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The rules generalized
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Examples
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Examples
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Case of base b
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Higher derivatives
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Application
One of the prime virtues of the logarithm is its
ability to convert a multiplication into an
addition and a division into a subtraction Example

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Contd
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Another example
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Optimal Timing
Application to Value of wine grows over time

• Problem when to sell the wine to maximize
  profit. Assumption no storage cost
• Need to discount each V to its present value.
• Interest rate has to be specified r

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Application to Timber Cutting
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Application of exponential and logarithmic
derivatives
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Examples
Find the rate of growth of

Find the rate of growth of
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Rate of growth Combination of functions
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Rate of growth Combination of functions
Example C grows at rate of a, H grows at rate of
ß
,
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Example 4 Exports GG(t) has a growth rate
a/t and export services SS(t) has a growth
rate b/t

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Finding Point Elasticity
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Example Find the point elasticity of the demand
function