Logarithmic Functions and Their Graphs

1. Definition of a Logarithmic Function

A logarithmic function is the inverse of an exponential function. It is defined as:
where:
  • is the base of the logarithm, with and .
  • (logarithms are only defined for positive values).
  • represents the exponent to which must be raised to produce .

Common Logarithms

Common Logarithm:

Properties of Logarithms

  • because .
  • because .
  • and (Inverse Properties).
  • If , then (One-to-One Property).

Natural Logarithms

The natural logarithm, denoted as , is particularly important in calculus and mathematical modeling. It is the logarithm to the base , a fundamental constant in mathematics that appears in growth and decay problems, probability, and many natural processes.
Natural Logarithm:
(base , where )

Properties of Natural Logarithms

  • because .
  • because .
  • and (Inverse Properties).
  • If , then (One-to-One Property).

Example Evaluations

  • because .
  • because .
  • because the natural logarithm and the exponential function are inverses, meaning since is exactly what must be raised to to get .

2. Graph of a Logarithmic Function

Key Features of

  • Domain: (Only positive values of are allowed).
  • Range: (Logarithms can take any real value).
  • Vertical Asymptote: (The graph approaches but never touches the -axis).
  • -intercept: since .
  • Increasing Function: The function is always increasing for .
  • Reflection Property: The graph of is the reflection of across the line .

Comparing Different Bases

  • If , the function increases slowly (e.g., is steeper than ).
  • If , the function decreases (e.g., is a decreasing function).
Example Graphs:
  • grows faster than .
  • behaves similarly but grows slightly faster than .

3. Transformations of Logarithmic Functions

A logarithmic function can be transformed using shifts, reflections, and stretches:
where:
  • affects vertical stretch/compression and reflection.
  • shifts the graph horizontally.
  • shifts the graph vertically.

Example Transformations

  1. shifts the graph units right.
  2. shifts the graph units up.
  3. reflects the graph across the -axis.
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4. Properties of Logarithms

Change-of-Base Formula

The change-of-base formula allows us to evaluate logarithms with any base:
For example, using base :

Properties for Expanding and Condensing Logarithms

  • Product Rule:
  • Quotient Rule:
  • Power Rule:

Example Expansions

Example Condensations


5. Applications of Logarithms

Logarithmic functions are used in various real-world applications:
  • Earthquake Magnitude : The Richter scale measures earthquake intensity.
  • pH Scale : Measures acidity based on hydrogen ion concentration.
  • Exponential Growth and Decay: Logarithms are used to solve for time in equations like .
  • Financial Modeling: Compound interest and investments use logarithmic functions to determine growth over time.
  • Sound Intensity (Decibels): The logarithmic scale measures sound levels.