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Magazine Chapter 12: Problem 80
Graph each logarithmic function. $$ f(x)=\log _{1 / 2} x $$
Short Answer
Expert verified
Sketch the curve passing through points (1, 0), (1/2, 1), and (2, -1), with a vertical asymptote at \( x = 0 \). The graph is decreasing.
Step by step solution
01
Understand the Problem
We are asked to graph the logarithmic function \( f(x) = \log_{1/2} x \). This function represents the logarithm of \( x \) with base \( \frac{1}{2} \), which is less than 1. Understanding the properties of logarithmic functions with bases less than 1 is crucial as they are decreasing functions.
02
Identify Key Characteristics
Identify the key characteristics of the log function \( f(x) = \log_{1/2} x \):- **Domain**: The domain is all positive real numbers, \( x > 0 \), because you cannot take the logarithm of zero or a negative number.- **Range**: The range is all real numbers because \( \log_{1/2} x \) can output any real number.- **Asymptote**: There is a vertical asymptote at \( x=0 \).- **Intercept**: The graph intersects the x-axis at \( (1, 0) \) because \( \log_{1/2} 1 = 0 \).
03
Evaluate at Key Points
Calculate a few values of \( f(x) \) at critical points:- \( f(1) = \log_{1/2}(1) = 0 \). This gives us the x-intercept.- \( f(1/2) = \log_{1/2}(1/2) = 1 \).- \( f(2) = \log_{1/2}(2) = -1 \).These points will help sketch the curve accurately.
04
Sketch the Graph
Plot the key points on the coordinate plane based on evaluations:- Point (1, 0)- Point (1/2, 1)- Point (2, -1)Draw the curve passing through these points, remembering that the function approaches but never touches the vertical asymptote at \( x=0 \), and it decreases as \( x \) increases because the base is less than 1.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Logarithmic Functions
Graphing a logarithmic function can initially seem daunting, but it's made simpler by focusing on key characteristics. In our example, we have the function \( f(x) = \log_{1/2} x \). Because the base, \( \frac{1}{2} \), is less than 1, this is a decreasing function. The graph of a logarithmic function is shaped like a curve that approaches a vertical asymptote without touching it.
Here's a step-by-step approach to graphing such functions:
Here's a step-by-step approach to graphing such functions:
- **Determine the domain:** Logarithmic functions are only defined for positive input values, meaning \( x > 0 \).
- **Identify the vertical asymptote:** This means the line the graph will approach but never touch, located at \( x = 0 \).
- **Locate intercepts:** For \( f(x) = \log_{1/2} x \), the graph will intercept the x-axis where \( x = 1 \).
- **Plot critical points:** Evaluate the function at specific values of \( x \) such as \( x = \frac{1}{2} \), \( x = 1 \), and \( x = 2 \) to note key coordinates.
- **Draw the curve:** Considering all the above points, you sketch a smooth, continuous curve that passes through known coordinates and trends towards the vertical asymptote.
Properties of Logarithmic Functions
Logarithmic functions unveil fascinating properties that help in understanding their behavior and applications. When discussing the function \( f(x) = \log_{1/2} x \), the base \( \frac{1}{2} \) particularly highlights certain attributes.
Here are the noteworthy properties of logarithmic functions:
Here are the noteworthy properties of logarithmic functions:
- **Domain and Range:** As logarithms can only operate on positive numbers, the domain is \( x > 0 \). Regardless of the base, the range consists of all real numbers.
- **Inverse Relation:** Logarithms are the inverse of exponential functions. For instance, \( y = a^x \) can be inverted as \( x = \log_a y \) and vice versa.
- **Base Implications:** The base governs the behavior of the function - if the base is greater than 1, the function is increasing; if it's between 0 and less than 1, the function is decreasing, like our example.
- **Zero Point:** A crucial property is that \( \log_a a = 1 \) and \( \log_a 1 = 0 \), illustrating why the graph of \( f(x) = \log_{1/2} x \) passes through \( (1, 0) \).
- **Vertical Asymptote:** Every logarithmic function approaches but never crosses \( x = 0 \).
Base of Logarithm
The base of a logarithm is foundational to understanding its function and graph. It affects the way the graph behaves.
In \( f(x) = \log_{1/2} x \), the base \( \frac{1}{2} \) is particularly significant because it is less than 1.
Here鈥檚 what different bases mean in terms of graph behavior:
In \( f(x) = \log_{1/2} x \), the base \( \frac{1}{2} \) is particularly significant because it is less than 1.
Here鈥檚 what different bases mean in terms of graph behavior:
- **Positive Base Greater than 1:** The function will be increasing, meaning as \( x \) increases, \( f(x) \) increases.
- **Positive Base Between 0 and 1:** Like the base \( \frac{1}{2} \), the function is decreasing. As \( x \) increases, \( f(x) \) decreases. This reflects negatively scaling, typical of decaying processes.
- **Base Equals 1:** Logarithmic functions do not exist because \( \, \log_1 x \) is not defined beyond 1 (\( a^{anything} = 1 \) only when \( anything = 0 \)).
