Objectives: In this section, we define the logarithmic function as the inverse of the exponential function. The properties of the logarithmic function are discussed. The graphs of the various logarithmic function and their geometrical transformations are investigated. After working through these materials, the student should be able

• to discuss the definition of the logarithmic function;
• to describe the properties and graphs of logarithmic functions;
• to recognize graphically and algebraically geometric transformations of the logarithmic functions.

Modules:

• First, you should recall the material on inverse functions.

 Definition. Let a be a positive real number. The logarithm with base a f(x) = loga(x). is the inverse of the exponential function g(x) = ax. The natural logarithm f(x) = ln(x). is the inverse of the exponential function g(x) = ex. In other words, ln(x) = loge(x).

• The domain of f(x) = loga(x) consists of all positive real numbers.
Discussion.

• The range of f(x) = loga(x) is the collection of all real numbers.
Discussion.

• loga(ax) = x and aloga(x) = x. In particular, ln(ex) = x and eln(x) = x.
Discussion.

• If a < 1 then f is a decreasing function and if a > 1 then f is an increasing function.
Discussion.

• Logarithmic functions satisfy the following properties:

1. loga(r s) = loga(r) + loga(s)
2. loga(r/s) = loga(r) - loga(s)
3. loga(rs) = s loga(r)

• Many calculators and computer programs do not have the logarithmic function loga(x) but do have the natural logarithm, ln(x); in this case, the following identity is useful:

loga(x) = ln(x) / ln(a)

• Examples:

• Use this LiveMath notebook to view an animation showing the graphs of a parametrized family of logarithmic functions.

• LiveMath notebooks to explore graphically and symbolically the effect of transforming logarithmic functions.

• A Java applet explore graphically the effect of transforming logarithmic functions symbolically.