Logarithmic Functions

1
Logarithmic Functions

• Just another way to write exponents

2
Definition, Domain Graphs

• Definition
• Relating Logarithms to Exponents
• Changing Exponential Expressions to Logarithmic
  Expressions and vice versa
• Finding the Exact Value of a Logarithmic
  Expression
• Domain
• Graphs
• Best Tutorial that Ive found

3
Properties of a Logarithmic Function

• The domain is the set of positive real numbers
  the range is all real numbers (inverse of
  exponential function)
• The x-intercept of the graph is 1. There is no
  y-intercept
• The y-axis (x 0) is a vertical asymptote of the
  graph
• A logarithmic function is decreasing if 0ltalt1 and
  increasing if a gt 1.

4
Properties of a Logarithmic Function

• The graph of f contains the points (1, 0),
• (a,1), and 1/a, -1).
• The graph is smooth and continuous, with no
  corners or gaps.

5
Natural Log Function

• Definition
• Y ln x if and only if x ex

6
Common Logarithm Function

• Definition
• Y log x if and only if x 10y
• Graphs using transformations

7
Logarithmic Equations

• Solve Change equation to
• exponential equation

8
Logarithmic Equation

• Solve

9
Logarithmic Equation

• Solve

10
Alcohol and Driving

• The concentration of alcohol in a persons blood
  is measurable. Recent medical research suggests
  that the risk R (given as a percent) of having an
  accident while driving a car can be modeled by
  the equation
• R 6ekt
• where x is the variable concentration of alcohol
  in the blood and k is a constant

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Alcohol and Driving

• Suppose that a concentration of alcohol in the
  blood of 0.04 results in a 10 risk (R 10) of
  an accident. Find the constant k in the equation.
  Graph the equation using this k.
• Using this value of k, what is the risk if the
  concentration is 0.17?
• Using the same value of k, what concentration of
  alcohol corresponds to a risk of 100?

12
Alcohol and Driving

• If the law asserts that anyone with a risk of
  having an accident of 20 or more should not have
  driving privileges, at what concentration of
  alcohol in the blood should a driver be arrested
  and charged with a DUI (Driving Under the
  Influence).

13
Healing of Wounds

• The normal healing of wounds can be modeled by an
  exponential function. If the formula describes
  the area of a wound on the nth day following an
  injury when no infection is present to retard the
  healing. Suppose that a wound initially had an
  area of 100 sq. mms.

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Healing of Wounds

• (a) If healing is taking place, how many days
  should pass before the wound is one-half its
  original size?
• (b) How long before the wound is 10 of its
  original size?

15
Additional Examples

• More Examples of Logarithmic Equations