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Chapter 13: Problem 73

Graph each logarithmic function. $$f(x)=\log _{3} x$$

Short Answer

Expert verified
To graph the function \(f(x)=\log_3{x}\), consider the following properties: - The function has a vertical asymptote at \(x=0\). - The domain is \(x>0\), and the range is all real numbers. - The graph passes through the point \((1, 0)\) and has an increasing curve that increases at a decreasing rate. With these characteristics, plot the function for \(f(x)=\log_3{x}\), focusing on the general shape and asymptotes.

Step by step solution

01

Identify the base and the function

The logarithmic function given is \(f(x)=\log _{3} x\). Here, the base is \(3\) and the function represents the power to which \(3\) must be raised to obtain \(x\). The input is \(x\), and the output is the function value \(f(x)\).
02

Determine the domain and range of the function

The domain of a logarithmic function is the set of all positive real numbers, because we cannot take the logarithm of a negative number or zero. Thus, the domain of \(f(x)=\log _{3} x\) is \(x > 0\). The range of a logarithmic function is all real numbers. This is because the logarithm can take any value, from negative infinity to positive infinity. So, the range of \(f(x)=\log _{3} x\) is \(-\infty < f(x) < \infty\).
03

Find the vertical asymptote

A vertical asymptote occurs when the function approaches infinity or negative infinity as the input approaches a specific value. In the case of a logarithmic function, there is a vertical asymptote at \(x=0\), because as the input approaches zero, the function's value approaches negative infinity.
04

Identify the general shape of the graph

The general shape of the graph of a logarithmic function is a curve that increases as the input increases. The function starts near the vertical asymptote at \(x=0\) and increases slowly, eventually passing through the point \((1, 0)\) since \(\log_{3}{1} = 0\). As the input increases, the graph of the function increases at a decreasing rate.
05

Plot the function

Given the properties identified above, the graph of the function \(f(x) = \log_3{x}\) should have the following characteristics: - A vertical asymptote at \(x = 0\) - A domain of \(x > 0\) - A range of all real numbers - A point at \((1, 0)\) - An increasing curve that increases at a decreasing rate Plot the function using these properties as a guide. Keep in mind that while the graph represents the general shape of the logarithmic function, it may not include specific points that would be obtained by evaluating the function for various inputs.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain and Range
When dealing with a logarithmic function like \( f(x) = \log_{3}{x} \), it's crucial to understand the domain and range. The **domain** of a logarithmic function, in general terms, consists of all positive real numbers. This is because logarithms are undefined for zero and negative numbers. Therefore, the domain of \( f(x) = \log_{3}{x} \) is \( x > 0 \).
On the other hand, the **range** of a logarithmic function is all real numbers. A logarithmic function's output can be any real value, varying from negative infinity to positive infinity. So, the range of \( f(x) = \log_{3}{x} \) is \(-\infty < f(x) < \infty\).
Vertical Asymptote
Logarithmic functions like \( f(x) = \log_{3}{x} \) inherently have what is known as a vertical asymptote. This represents a line that the graph of the function will get infinitely close to but never actually touch. For logarithmic functions with the form \( \log_{b}{x} \), the vertical asymptote is located at \( x = 0 \).
As \( x \) approaches zero from the positive side, the value of the function \( f(x) \) approaches negative infinity. Essentially, this means that the graph "dives downwards" as it nears \( x = 0 \). The key characteristic of a vertical asymptote in logarithmic functions is how it shapes the initial behavior of the graph.
Graphing Functions
Graphing a logarithmic function such as \( f(x) = \log_{3}{x} \) consists of plotting key characteristics and points. Firstly, recognize the curve begins near the vertical asymptote at \( x = 0 \), and as \( x \) increases, the curve rises towards the right.
Key points to note during graphing:
  • The graph passes through \((1, 0)\) because \(\log_{3}{1} = 0\).
  • The curve increases at a decreasing rate.
  • It remains in the first quadrant of the coordinate plane.
Remember, this pattern is typical for all logarithmic functions regardless of the base, while specific values shift based on that base.
Base of Logarithms
Logarithmic functions may vary in base, which influences their graph's shape and growth pattern. In \( f(x) = \log_{3}{x} \), the base is 3. This tells us how the function increases as \( x \) increases.
The value of the base affects how steep or shallow the curve is. A larger base (e.g., 10) leads to a graph that increases more slowly as \( x \) grows, whereas a smaller base (e.g., 2) would result in a steeper graph. Whatever the base is, the overall shape remains consistent:
  • Starts near a vertical asymptote.
  • Slowly rises through point \((1, 0)\).
  • Continues increasing but at a slower rate.
Understanding the base gives insight into how the logarithmic function behaves and scales.

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Most popular questions from this chapter

In \(1995,\) the population of a rural town in Kansas was \(1682 .\) The population is decreasing at a rate of \(0.8 \%\) per year. Use \(y=y_{0} e^{-0.0088}\) to answer the following questions. a) What was the population of the town in \(2000 ?\) b) In what year would it be expected that the population of the town is \(1000 ?\)

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