Dealing with Logs

Dealing with Logs

The Logarithmic Function

• Definition: For positive real numbers x and b (where b > 0), the expression $y = ext{log}_b x$ is known as the logarithmic function with base b.

  • Here,

  • $y$ is referred to as the logarithm.

  • $b$ is called the base of the logarithm.

  • $x$ is referred to as the argument of the logarithm.

• Equivalence: The equations $y = ext{log}_b x$ and $b^y = x$ define the same relationship between x and y.

  • The expression $y = ext{log}_b x$ is referred to as the logarithmic form.

  • The expression $b^y = x$ is referred to as the exponential form.

Properties of the Logarithmic Function

• Function Definition: The logarithmic function defined by
  $f(x) = ext{log}_b x$ where (b > 0 and b ≠ 1) exhibits the following properties:

1. Increasing/Decreasing Behavior:

   • If b > 1 , the function $f$ is an increasing function.

   • If 0 < b < 1 , the function $f$ is a decreasing function.

2. Domain:

   • The domain of the logarithmic function is given by the interval:
     (0, \, +)

3. Range:

   • The range of the logarithmic function includes all real numbers:
     (-, \, +)

4. Asymptotic Behavior:

   • The line $x = 0$ (the y-axis) serves as a vertical asymptote of the function.

5. Key Points on the Graph:

   • The function crosses through the point
     $(1, 0)$ because
     $f(1) = ext{log}_b(1) = 0$.

Graph of the Logarithmic Function ( f(x) = ext{log}_b x )

• Specific Example: When dealing specifically with the base 2 logarithm,
  the function can be represented as:
  $y = ext{log}_2 x$

• Summary of Critical Values:

  • Domain:
    (0, \, +)

  • Range:
    (-, \, +)

  • Important Point:
    $(1, 0)$

  • Horizontal Asymptote:
    $x = 0$