Introduction to Logarithmic Functions

Overview

A logarithmic function is the inverse of an exponential function. If an exponential function is written as:

\( f(x) = a \cdot b^x \)

The corresponding logarithmic function is written as:

\( f(x) = \log_b(x) \)

Where \( b \) is the base, and \( x \) is the argument of the logarithm. The base \( b \) must be greater than 0, and \( b \neq 1 \).

Key Properties of Logarithmic Functions

  • Domain: The domain of a logarithmic function is all positive real numbers (\( x > 0 \)), since the argument \( x \) must be positive.
  • Range: The range of a logarithmic function is all real numbers (\( -\infty, \infty \)).
  • Asymptote: Logarithmic functions have a vertical asymptote at \( x = 0 \), which means the graph approaches but never crosses the y-axis.
  • Increasing or Decreasing: If the base \( b > 1 \), the logarithmic function is increasing, and if \( 0 < b < 1 \), the function is decreasing.

Example 1: Logarithmic Growth

Consider the function:

\( f(x) = \log_2(x) \)

This is an example of a logarithmic function with base 2. It is an increasing function, and the graph will approach the vertical asymptote at \( x = 0 \) and increase as \( x \) gets larger. The domain is \( x > 0 \), and the range is all real numbers.

Example 2: Logarithmic Decay

Consider the function:

\( f(x) = \log_{0.5}(x) \)

This is an example of a logarithmic function with base 0.5. Since \( b = 0.5 < 1 \), this is a decreasing function. The graph approaches the vertical asymptote at \( x = 0 \) but decreases as \( x \) increases.

Practice Questions

  1. Question 1: Identify the domain and range of the function \( f(x) = \log_3(x) \).
    Solution

    The domain of the function is \( x > 0 \), and the range is all real numbers (\( -\infty, \infty \)).

  2. Question 2: For the function \( f(x) = \log_5(x) \), what is the end behavior as \( x \to 0^+ \) and as \( x \to \infty \)?
    Solution

    As \( x \to 0^+ \), \( f(x) \to -\infty \). As \( x \to \infty \), \( f(x) \to \infty \).

  3. Question 3: What is the vertical asymptote of the function \( f(x) = \log_7(x) \)?
    Solution

    The vertical asymptote is at \( x = 0 \), because logarithmic functions approach the y-axis but never cross it.

  4. Question 4: Write the logarithmic form of the exponential equation \( 7^x = 49 \).
    Solution

    The logarithmic form is:

    \( x = \log_7(49) \)

  5. Question 5: Given the function \( f(x) = \log_2(x) \), find \( f(8) \).
    Solution

    We have:

    \( f(8) = \log_2(8) = 3 \)

    Because \( 2^3 = 8 \).

  6. Question 6: Sketch the graph of \( f(x) = \log_4(x) \). Label the asymptote and describe the behavior of the function.
    Solution

    The graph has a vertical asymptote at \( x = 0 \) and increases as \( x \) increases, passing through the point \( (1, 0) \) because \( \log_4(1) = 0 \).

→Properties of Logarithmic Functions