Exponential and Logarithmic Functions

1
Exponential and Logarithmic Functions
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Learning Objectives

• Exponential functions.
• Properties of exponentials.
• Logarithmic functions.
• Properties of logarithms.
• Exponential and logarithmic equations.
• Applications.

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Exponential Functions

• Exponential function an exponential function is
  of the form f(x) bx where bgt0, b ? 1, and x is
  any real number.
• Domain R.
• Range (0, ?).
• Euler number e 2.71828 is the base of the
  natural exponential function ex.
• Growth function bx where b gt 1.
• Decay function bx where 0 lt b lt 1.

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Exponential Functions
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Exponential Functions

• Example f(x) -2 . 3x1 .

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Properties of Exponentials

• For a,b gt0, b ? 1 an bn ? a b
• an am ? n m
• The exponential is a one-to-one function.

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Logarithmic Functions

• Logarithmic function the logarithmic function is
  the inverse of the exponential function.
• Logarithmic function of base b f(x) logbx ,
  for b ? 1.
• f(x) logbx ? f-1(x) bx where b ? 1, and x is
  any real number.
• a logbc ? ba c, where b ? 1.
• Domain (0, ?) (the range of exp).
• Range R (the domain of exp).

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Logarithmic Functions

• General log the general log is logbx . In
  Maple logb(x).
• Common log the common log is log10x, or log
  x.In Maple log10(x), log10(x).
• Natural log the general log is logex ln x.In
  Maple ln(x), logexp(1)(x).

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Logarithmic Functions
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Logarithmic Functions
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Logarithmic Functions

• Example f(x) -2 . log3(x1) .

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Properties of Logarithms

• For a,b gt0, b ? 1 logax logbx ? a b
• logan logam ? n m
• The logarithm is a one-to-one function. logbbx
  x
• b logb x x
• logb1 0
• logbx ln(x) / ln(b)

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Properties of Logarithms

• logb (ac) logb a logb c
• logb (a/c) logb a - logb c
• logb (ac) c logb a
• logb (a) logc a / logc b

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Properties of Exponentials and Logarithms

• y logax ? ay x
• ay x ? y logax
• ax ex ln a

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Exponential and Logarithmic Equations

• Solve 85x1 182x-3 e ln (8) (5x1) eln(18)
  (2x-3)
• ln(8) (5x1) ln(18) (2x-3)x (5ln8 2ln18)
  -3ln18 ln8)x - (3ln18 ln8) / (5ln8
  2ln18)

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Exponential and Logarithmic Equations

• Solve log2 8 log2 9 logx 3 log2 (8 . 9)
  logx 3
• ln (72) / ln 2 ln3 / ln x
• ln x ln 3 . ln 2 / ln 72x e (ln 3 . ln 2
  / ln 72) 3 ln 2 / ln 2.36 3 ln 2 / ln
  2.2.2.3.3