Precalculus A Fuctional Approach to Graphing and Problem Solving

Section 4.5: Logarithmic Functions
Essential Ideas
  • For positive b and A, b not equal to 1,
               x = logbA     means      b x = A
    x is the called the logarithm and A is called the argument.

  • Exponential Property of Equality
    For positive real numbers b (b not equal to 1)
               b x = b y    if and only if     x = y

  • Logarithmic Notations
    A common logarithm is base 10: log x means log10x
    A natural logarithm is base e :      ln x means logex

  • Change of Base Theorem
                               logax = logbx / logbA

  • The function f defined for positive real numbers by
                               f (x) = logbx
    where b > 0, b not equal to 1, is called the logarithmic function with
    base b.

  • See the directory of curves for the graph of the logarithmic function

  • Properties of Logarithms (Grant's Tomb Properties)
                 logbbx = x   The exponent on a base b which is b x is x. Duh.. .
                 b logbx = x   logbx is the exponent of a base b which gives x.
                                     Double duh.... this is the definition of exponent.
                                     See why these are called Grant's Tomb properties?
  • The exponential and logarithmic functions with base b are inverse functions of one another.

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© 2011 Karl J. Smith. All rights reserved.