Question

Graph each function using a graphing utility and the Change-of-Base Formula. \(y=\log _{x+2}(x-2)\)

Short answer

Rewrite \(y = \log_{x+2}(x-2)\) using Change-of-Base formula. Domain is \(x > 2\). Graph within this domain.

Step-by-step solution

Rewrite the Logarithm Using Change-of-Base Formula

The Change-of-Base Formula allows rewriting a logarithm of any base in terms of natural or common logarithms. For the given function, rewrite \(y = \log_{x+2}(x-2)\) using the formula: \[y = \dfrac{\log(x-2)}{\log(x+2)}\]

Identify Restrictions on the Domain

To find the domain, ensure the arguments of the logarithms are positive and the denominator is not zero:1. \(x - 2 > 0\) implies \(x > 2\)2. \(x + 2 > 0\) implies \(x > -2\)Therefore, the domain is \(x > 2\).

Plot a Few Points

Choose a few values of \(x\) greater than 2 and calculate corresponding \(y\)-values. For example, take \(x = 3\):\[y = \dfrac{\log(3-2)}{\log(3+2)} = \dfrac{\log(1)}{\log(5)} = 0\]Repeat for other values (4, 5, etc.).

Use a Graphing Utility

Enter the equation \(y = \dfrac{\log(x-2)}{\log(x+2)}\) into a graphing calculator or software. Verify the plotted points match the calculated points and observe the graph's behavior.

Analyze the Graph

Examine the graph further for characteristics such as asymptotes, intercepts, and general shape. Notice how the function behaves as \(x\) approaches 2 from the right since this defines the lower boundary of the domain.

Key concepts

change-of-base formula

The Change-of-Base Formula is a powerful tool in logarithms. It allows us to convert a logarithm of any base to a fraction of logarithms with other bases—usually common logarithms (base 10) or natural logarithms (base e). This is especially helpful when dealing with logarithms that our calculators or graphing utilities cannot handle. For our function, we have the logarithm \(y = \log_{x+2}(x-2)\). Using the change-of-base formula, we can rewrite it as \[y = \dfrac{\log(x-2)}{\log(x+2)}\]. This conversion simplifies our work significantly, enabling easier calculation and graphing.

domain restrictions

Domain restrictions are crucial when working with logarithmic functions. They tell us which values of \(x\) we can use. For a logarithmic equation like \(y = \dfrac{\log(x-2)}{\log(x+2)}\), each argument inside the logarithms must be positive, and the denominator must not be zero. Let's break this down:
- \(x-2 > 0\), so \(x > 2\).
- \(x+2 > 0\), so \(x > -2\).
However, because the denominator cannot be zero, we stick with \(x > 2\).
So, the domain for our function is \(x > 2\). This means we can only input values of \(x\) that are greater than 2.

graphing techniques

Graphing logarithmic functions can be tricky, but some techniques make it easier. To graph \(y = \dfrac{\log(x-2)}{\log(x+2)}\), follow these steps:
- **Choose Points:** Select values of \(x\) greater than 2. For instance, start with x=3, calculate \(y = \dfrac{\log(1)}{\log(5)} = 0\). Repeat this for other values like 4, 5, etc.
- **Graphing Utility:** Use a graphing calculator or software to input the equation. Enter \(y = \dfrac{\log(x-2)}{\log(x+2)}\) and plot the graph.
- **Verify Points:** Compare the calculated points with the plotted ones to make sure they match.
- **Analyze:** Look at the graph to identify intercepts, asymptotes, and the shape. Watch how the graph behaves near the domain restriction (\(x>2\).