Chiang Ch' 10 Exponential and Logarithmic Functions

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Chiang Ch. 10 Exponential and Logarithmic
Functions

• 10.1 The Nature of Exponential Functions
• 10.2 Natural Exponential Functions and the
  Problem of Growth
• 10.3 Logarithms
• 10.4 Logarithmic Functions
• 10.5 Derivatives of Exponential and Logarithmic
  Functions
• 10.6 Optimal Timing
• 10.7 Further Applications of Exponential and
  Logarithmic Derivatives

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10.1(a) Types of functions (Chiang, p.26)
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7.3.2 Inverse-function rule

• This property of one to one mapping is unique to
  the class of functions known as monotonic
  functions
• Definition of a function (p. 17)
• Function one y for each x and
• Monotonic function one x for each y
• One x for each y, aka inverse function

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7.3 Rules of Differentiation Involving Functions
of Different Variables
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10.1 The Nature of Exponential Functions

• 10.1(a) Simple exponential function
• 10.1(b) Graphical form
• 10.1(c) Generalized exponential function
• 10.1(d) A preferred base

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10.1(a) Simple exponential function

• y f(x) bx
• where base b gt 1, x is exponent, f(x)? ?
• The term exponent (x) refers to the power to
  which a base number (b) is raised.
• Base exclusions
• b ? 1 and b ? 0, because f(x) 1x 1 f(x)
  0x 0, i.e., constants
• 0 lt b lt 1 excluded since they can be expressed as
  negative exponents
• blt0 excluded because many values of f(x) from the
  domain would be imaginary, e.g., (-b)½
• popular bases e and 10

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10.1(c) Generalized exponential function

• Where
• y dependent variable
• b base
• t independent variable
• a vertical scale factor (directly related)
• c horizontal scale factor (inversely related)

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10.1(b) Graphical Exponential Functions ybt
where b3,e,2
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10.1(d) A preferred base (e)
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y1e and y2(11/m)m as m??
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10.5 Derivative of the inverse of yet, i.e. yln
t (p. 277)
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10.5 Derivative of yet using the Inverse
function rule
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10.5(d) The case of base b
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10.1(d) A preferred base (e) (e) 2.71828,
irrational number w/ beautifully simple
characteristics
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10.1(d) A preferred base (e) (e) 2.71828,
irrational number w/ beautifully simple
characteristics)
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10.1(b) Graphic for f(x)ex
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10.1(d) A preferred base (e) (e) 2.71828,
irrational number w/ beautifully simple
characteristics)
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10.1(d) A preferred base (e) (e) 2.71828, an
irrational number w/ beautifully simple
characteristics)
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10.2 Natural Exponential Functions and the
Problem of Growth

• 10.2(a) The number e
• 10.2(b) An economic interpretation of e
• 10.2(c) Interest compounding and the function
  Aert
• 10.2(d) Instantaneous rate of growth
• 10.2(e) Continuous vs. discrete growth
• 10.2(f) Discounting and negative growth

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10.2(a) The number e
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10.2(bc) Interest compounding and the function
Aert

• y Aert is the value of an A investment at
  nominal interest rate r / compounding period
  compounded t times over the investment period (
  days, months, or years) (growth in an
  investment)
• As m ? ?, e ? 2.71828
• 1 initial investment
• 1 100 return
• (if m 1 at the end of the year, then e 2
• if m gt 1, then r 100 is a nominal rate and e
  gt 2)
• .71828 interest on the interest received
  during the year as m ? ?, r 171.8 (effective
  rate)

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10.2(e) Continuous vs. discrete growth
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10.2(e) Continuous vs. discrete growth
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Relation between effective (i) and nominal (r)
rates of interest
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10.2(d) Instantaneous rate of growth of a stock
at two points in time (Chiang, pp. 278-289)
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10.2(d) Instantaneous rate of growth of a stock
at two points in time (Chiang, pp. 278-289)
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10.2(f) Discounting as negative growth or rate of
decay

• future V present A f(compounding the
  present A) f(discounting the future V)

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10.3 Logarithms

• 10.3(a) The meaning of logarithm
• 10.3(b) Common log and natural log
• 10.3(c) Rules of logarithms
• 10.3(d) An application

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10.3(a) The meaning of logarithm

• Exponents Common logs
• (solve for y given t) (solve for t given y)

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10.3(b) Common log and natural log

• Exponent Natural log

• Exponent Common log

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10.3(c) Rules of logarithms in the land of
exponents

• Product
• Quotient
• Power
• Base inversion
• Base conversion

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10.4 Logarithmic Functions

• 10.4(a) Log functions and exponential functions
• 10.4(b) The graphical form
• 10.4(c) Base conversion

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10.4(a) Log functions and exponential functions

• Exponential function Log function
• Dependent variable on left
• Monotonically increasing functions
• If ln y1 ln y2, then y1 y2

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10.4(b) The graphical formyet (blue), y2t
(red-top), yln(t) (red), 45o (green)
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10.4(c) Base conversion

• Let er bc
• Then ln er ln bc
• r ln bc c ln b
• Therefore
• er e c ln b
• And
• y Abct Ae(c ln b)t Aert

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10.5 Derivatives of Exponential and Logarithmic
Functions

• 10.5(a) Log-function rule
• 10.5(b) Exponential-function rule
• 10.5(c) The rules generalized
• 10.5(d) The case of base b
• 10.5(e) Higher derivatives

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10.5 Derivative of yln t
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10.5(a) Log-function rule

• Derivative of a log function with base e
• Simple General

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10.5(b) Exponential-function rule

• Derivative of an exponential function with base e
• Simple General

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10.3(d) An application

• The common use of the logarithmic transformation
  in economic research is when estimating
  production functions and other multiplication
  nonlinear theoretical specifications
• Transforming a multiplicative production function
  into a logarithmic one suitable for estimation by
  linear regression. Let Q output and L K are
  labor and capital inputs respectively

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Total differential of Q using logs
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Exponent example
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10.5(d) The case of base b
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10.5(e) Higher derivatives
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10.6 Optimal Timing

• 10.6(a) A problem of wine storage
• 10.6(b) Maximization conditions
• 10.6(c) A problem of timber cutting

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10.6(a) A problem of wine storage
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10.6(a) A problem of wine storage
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10.6(b) Maximization conditions
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10.6(a) A problem of wine storage plot of
.5(t)-.5r, r.10, t25
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10.6(c) Timber cutting problemplot of
.5(t)-.5ln(2)r, r.05, t48
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10.7 Further Applications of Exponential and
Logarithmic Derivatives

• 10.7(a) Finding the rate of growth
• 10.7(b) Rate of growth of a combination of
  functions
• 10.7(c) Finding the point elasticity

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10.7(a) Finding the rate of growth
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10.7(b) Rate of growth of a combination of
functions Example 3 consumption pop.
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10.7(b) Rate of growth of a combination of
functions Example 3 consumption pop.
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10.7(c) Finding the point elasticity
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