Properties of Logarithmic Functions
Overview
Logarithmic functions are the inverse of exponential functions. The general form of a logarithmic function is:
\( f(x) = \log_b(x) \)
Where \( b \) is the base and \( x \) is the argument. In this section, we will explore the key properties of logarithmic functions.
Key Properties of Logarithmic Functions
- Domain: The domain of the logarithmic function is all positive real numbers, i.e., \( x > 0 \). The argument of the logarithm must be positive.
- Range: The range of a logarithmic function is all real numbers, i.e., \( (- \infty, \infty) \), because logarithmic functions can produce any real number as an output.
- Vertical Asymptote: The graph of a logarithmic function has a vertical asymptote at \( x = 0 \). As \( x \) approaches 0 from the right, \( f(x) \) tends to negative infinity.
- Monotonicity: If the base \( b > 1 \), the function is increasing. If \( 0 < b < 1 \), the function is decreasing. Logarithmic functions are always continuous and do not have any gaps or breaks.
- Intercept: A logarithmic function always passes through the point \( (1, 0) \), because \( \log_b(1) = 0 \) for any base \( b \).
- Behavior as \( x \to 0^+ \): As \( x \) approaches 0 from the positive side, the logarithmic function tends to negative infinity. The graph gets closer and closer to the vertical asymptote but never crosses it.
- Behavior as \( x \to \infty \): As \( x \) increases towards infinity, the logarithmic function increases indefinitely (if \( b > 1 \)) or decreases indefinitely (if \( 0 < b < 1 \)), but the rate of increase or decrease slows down as \( x \) gets larger.
Examples of Logarithmic Functions
Example 1: \( f(x) = \log_2(x) \)
This is a logarithmic function with base 2. Its graph is increasing, and it passes through the point \( (1, 0) \). It has a vertical asymptote at \( x = 0 \), and as \( x \) approaches infinity, \( f(x) \) increases without bound.
Example 2: \( f(x) = \log_{0.5}(x) \)
This is a logarithmic function with base 0.5. Since the base is less than 1, the function is decreasing. It has a vertical asymptote at \( x = 0 \), and as \( x \) approaches infinity, \( f(x) \) decreases without bound.
Example 3: \( f(x) = \log_{10}(x) \)
This is a common logarithmic function, also known as the base-10 logarithm. It behaves similarly to other logarithmic functions with base greater than 1, increasing as \( x \) increases and approaching negative infinity as \( x \) approaches zero.
Practice Questions
- Question 1: Determine the domain and range of the function \( f(x) = \log_3(x) \).
Solution
The domain of \( f(x) \) is \( x > 0 \) because the argument of the logarithm must be positive. The range is all real numbers, \( (- \infty, \infty) \), because the logarithmic function can produce any real number.
- Question 2: What is the vertical asymptote of the function \( f(x) = \log_5(x) \)?
Solution
The vertical asymptote of the function is at \( x = 0 \), because logarithmic functions approach but never cross the y-axis.
- Question 3: Sketch the graph of \( f(x) = \log_2(x) \). Label the asymptote and the intercept.
Solution
The graph of \( f(x) = \log_2(x) \) has a vertical asymptote at \( x = 0 \) and passes through the point \( (1, 0) \). The graph increases as \( x \) increases and approaches negative infinity as \( x \) approaches 0 from the right.
- Question 4: For the function \( f(x) = \log_4(x) \), determine the behavior as \( x \) approaches 0 from the positive side and as \( x \) approaches infinity.
Solution
As \( x \to 0^+ \), \( f(x) \to -\infty \), and as \( x \to \infty \), \( f(x) \to \infty \).
- Question 5: What is the intercept of the function \( f(x) = \log_7(x) \)?
Solution
The intercept is at the point \( (1, 0) \), because \( \log_b(1) = 0 \) for any base \( b \).
- Question 6: For the function \( f(x) = \log_2(x) \), evaluate \( f(16) \).
Solution
We have:
\( f(16) = \log_2(16) = 4 \)
Because \( 2^4 = 16 \).