Problem 82 - College Algebra Essentials

Question

Use a graphing utility and the change-of-base property to graph each function. \(y=\log _{3}(x-2)\)

Step-by-Step Solution

Answer

After applying the change-of-base property, we get the transformed function as \(y = \frac{\log (x-2)}{\log 3}\). This function is then graphed, which gives an upward sloping curve that begins at the point (2,0) and increases logarithmically as \(x\) progresses past 2. No values exist when \(x\) is less than or equal to 2.

Step 1: Apply the change-of-base property

In order to graph the function using a graphing utility, we first need to apply the change-of-base property. We convert the base 3 logarithm to base 10 (since it's easy to work with). So, our function \(y = \log _{3}(x-2)\) becomes \(y = \frac{\log (x-2)}{\log 3}\).

Step 2: Graph the function

Now, with the function in terms of base-10 logarithms, graph the function using a graphing utility by inputting the function string \(y = \frac{\log (x-2)}{\log 3}\) into it. Any graphing utility can be used to do this, such as Desmos or GeoGebra.

Step 3: Interpret the graph

After successfully plotting the function graph, this will give an upward sloping curve that begins at the point (2,0), indicating that this is a logarithmic function shifted two units to the right. As \(x\) progresses past 2, the value of \(y\) will slowly increase, indicating that y varies logarithmically with \(x\). Additionally, no values will exist for \(x\) less than or equal to 2, due to the nature of logarithmic functions.

Key Concepts

Change-of-Base Property

When working with logarithms, especially for graphing purposes using utilities, the change-of-base property becomes quite handy. The change-of-base property allows you to convert a logarithm with any base to a logarithm with a different base, typically base 10 or base e. This simplifies calculations, especially when using technology that is designed to work more seamlessly with these bases.
For example, consider our original function:

• Original: \( y = \log_3(x-2) \)
• Using Change-of-Base: \( y = \frac{\log(x-2)}{\log 3} \)

This transformation utilizes basic properties of logarithms which state that:

• \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)

Where \( b \) is the base you want to convert from, and \( c \) is the base you are converting to, like 10 or e.

Logarithmic Functions

Logarithmic functions play a critical role when dealing with exponential relationships and data that vary multiplicatively. The general form of a logarithmic function is \( y = \log_b(x) \), where \( b \) is the base, and \( x \) is the value you input. A base that is greater than 1 indicates that as \( x \) increases, \( y \) will also increase. For negative values of \( x \) or zero, the function is undefined because logarithms are not defined for non-positive numbers. In the context of our function \( y = \log_3(x-2) \), this is a logarithmic function with a specific base and a horizontal shift:

• The base of 3 indicates growth.
• The term \( (x-2) \) shows that the function is shifted 2 units to the right, meaning the function only takes inputs when \( x > 2 \).

This shift is crucial as it changes where the function "starts" on an \( x \)-axis and helps in locating points to plot.

Graph Interpretation

Interpreting the graph of a logarithmic function provides insight into how the function behaves under different conditions. With graphs, you can observe points, intercepts, and the general increasing or decreasing pattern of the function. Once you have the graph plotted, the logarithmic curve typically has a specific shape: For \( y = \frac{\log(x-2)}{\log 3} \), you will observe:

• The curve starts at \( (2,0) \), which is known as the vertical asymptote where the function approaches but never touches \( x=2 \).
• The function is undefined for \( x \leq 2 \), illustrating an important characteristic of logarithms: they can't take non-positive numbers as inputs.
• Past \( x = 2 \), \( y \) increases slowly, showing the typical behavior of a logarithm where changes are rapid initially but slow down as \( x \) increases.

By using a graphing utility, students can see this behavior visualized, aiding in understanding the concept and its real-world applications, be it in science, economics, or other fields where growth and multiplicative factors are relevant.