Problem 43 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the graph of the equation. $$y=\log _{3} x$$

Step-by-Step Solution

Answer

To sketch the graph of \(y = \log_{3}x\), consider the following key characteristics: 1. There is a vertical asymptote at x=0. 2. The x-intercept occurs at x=1. 3. There is no y-intercept. 4. The graph increases as x > 1. Plot the points (1,0) and (3,1), along with the asymptote at x=0. The resulting graph should be a smooth curve, increasing as x becomes larger and never touching the x=0 line.

Step 1: Understanding the characteristics of logarithmic functions

A logarithmic function generally has the form \(y = \log_{b} x\), where b > 0 and \(b \neq 1\). The graph of the function will have a vertical asymptote at \(x = 0\) and will increase as x becomes larger, intersecting the y-axis at x = 1. It's important to note that as \(x\) approaches \(0\), the y values become negative, but never actually reach \(0\).

Step 2: Finding the x-intercept

To find the x-intercept, we need to find the value of x when y = 0. By plugging y = 0 into the equation: \(0 = \log_{3} x\) To rewrite this in exponential form, we will use the property \(y=\log_{b} x \) which can also be written \ \(b^y=x \) \(3^0 = x\) The x-intercept is therefore \(x = 1\).

Step 3: Finding the y-intercept

To find the y-intercept, we need to find the value of y when x = 0. However, the logarithmic functions' domain does not include x = 0, so there is no y-intercept.

Step 4: Plotting some other points

To help us sketch the graph, let's find the value of y at x = 3: \(y = \log_{3} 3 \) Since \(3^1 = 3\), the logarithm will be equal to 1: \(y = 1\) So, when \(x = 3\), \(y = 1\).

Step 5: Sketching the graph

Now that we have some points to plot on our graph, we can create a rough sketch that represents the characteristics of the logarithmic function: 1. Asymptote at x = 0 2. X-intercept at x = 1 3. Increasing graph when x > 1 4. Plot the point (1, 0) and (3, 1) Combine these characteristics to sketch the graph. Remember that the graph will not touch the x = 0 line since that is where the asymptote lies.

Key Concepts

Graph Sketching

When sketching the graph of a logarithmic function like \(y = \log_{3} x\), it's all about understanding its basic shape and properties. First, note the curve is only defined for values of \(x > 0\); this is because you cannot take the logarithm of a non-positive number. The function is continuous and increases slowly as \(x\) grows larger. For a sketch:

• Start by drawing a vertical asymptote line at \(x = 0\)
• Make note of important points such as the x-intercept when \(x = 1\)
• Include other points like \( (3,1) \) which help with representation
• Finally, show the increasing nature of the curve as x becomes larger

Use these steps to create a smooth curve that's close to the x-axis on the left and rises gradually on the right.

X-Intercept

To find where the graph of \(y = \log_{3} x\) crosses the x-axis, we need to determine the x-intercept. This is the point where the value of \(y\) is zero. This can be simplified using the equation:\[0 = \log_{3} x\]Converting this logarithmic equation into its exponential form gives:\[3^0 = x\]Thus, \(x = 1\). This tells us that when \(y\) equals zero, \(x\) is 1, indicating the graph crosses the x-axis at this point. The x-intercept is a foundational point for sketching the function and showing its behavior at \(x = 1\).

Vertical Asymptote

The vertical asymptote is a key feature of any logarithmic graph, including \(y = \log_{3} x\). It acts as a boundary that the graph approaches but never crosses. For the function \(y = \log_{3} x\), this vertical asymptote is located at \(x = 0\). Why is there an asymptote at \(x = 0\)?

• Because logarithmic functions are undefined for non-positive numbers.
• As \(x\) approaches 0 from the positive side, the y-values sharply decrease.

Remember, when sketching this function, the graph nears the line \(x = 0\) but never touches or crosses it.

Graph Characteristics

Understanding the graph characteristics of \(y = \log_{3} x\) can enhance your sketch and comprehension of logarithmic behaviors. Here are some core aspects to consider:- **Domain**: The function is only defined for \(x > 0\). There are no values of \(y\) when \(x\) is zero or negative.- **Range**: Since the graph can extend indefinitely upward or downward based on \(x\), the range is all real numbers.- **Curve Behavior**: - It rises gently but steadily as \(x\) increases beyond 1. - It never touches or crosses the vertical asymptote.- **Sympathetic Points**: - The graph is symmetric in that it is continuously increasing.These characteristics together allow the depiction of a typical logarithmic curve, maintaining the essential properties within its unbounded domain and range.