Exponential and Logarithmic Functions

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Chapter 4

• Exponential and Logarithmic Functions

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Example of Exponential Curve

• Cost of Computing is Declining

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Exponential Functions

• f(x) akx
• a constant
• k non-zero constant
• Exponential function with base a

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Example

• If a 10 and k 1
• f(0) 100 1
• f(1) 101 10
• f(2) 102 100
• f(3) 103 1000
• f(4) 104 10000

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Figure 4.1
f(x) 2x
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Figure 4.3
g(x) 2-x
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Figure 4.6
f(x) 3-x and 31-x
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Figure 4.8
f(x) 2-xx
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Example

• If current trends of burning fossil fuel and
  deforestation continue, then the increase in the
  amounts of atmospheric carbon dioxide in parts
  per million will be given by
• f(t) 375e.00609t

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• What will be the amount of carbon dioxide in 2000
  (t 0)?
• What will be the amount of carbon dioxide in 2025
  (t 25)?

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Example
e exp( )
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Group Work
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Number 13
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Number 15
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Number 17
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Applications of Exponential Functions

• Exponential growth function
• f(t) y0ekt
• f(t) y0bt

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Example

• Radioactive lead-210 decays to polonium-210. The
  amount y of radioactive lead-210 at time t is
  given by
• y y0e-.032t
• where t is time in years. How much of an
  initial 500 grams of lead-210 will remain after 5
  years?

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• y y0e-.032t
• 500exp(-.0325)
• 426 grams

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Example 1

• For health reasons, cigarette consumption in the
  United States has been decreasing for several
  decades. The number of cigarettes consumed (in
  millions) in year t is approximated by the
  exponential function
• C(t) y0e-.018654t
• where t is time in years and t0 corresponds
  to 1980.

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• a. If 635.5 million cigarettes were consumed in
  1980, what was cigarette consumption in 2005?
• C(t) y0e-.018654t
• First, determine y0.
• Second, calculate C(t) for t 25 (corresponding
  to 25 years since 1980).

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• C(t) y0e-.018654t
• When t 0, C(0) y0
• 635.5 million
• C(25) (635.5)e-.018654(25)
• 398.64 million cigarettes

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• b. If this trend continues, what will cigarette
  consumption be in 2012?
• t 32 for year 2012 (2012-1980)
• C(32) (635.5)e-.018654(32)
• 349.84 million cigarettes

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Example 2

• When money is placed in a bank account that pays
  compound interest, the amount in the account
  grows exponentially. Suppose an account grows
  from 1000 to 1316 in seven years.

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• a. Find a growth function of the form
• f(t) y0bt
• that gives the amount in the account at time
  t years.