Problem 79 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

We use the change-of-base property of logarithms to render the given function in terms of a base that's easier to work with, rewriting \(y=\log _{3} x\) as \(y = \frac{\log x}{\log 3}\). The graph will be the same for either equation.

Step 1: Identify the Function

The given function is \(y=\log _{3} x\). We need to graph this function using the change-of-base property of logs.

Step 2: Apply the Change-of-Base Property

The change-of-base property allows us to rewrite the logarithm in terms of a base that is more comfortable to work with. In this case, we'll rewrite it in terms of a base 10 logarithm. Using the property \(\log_b a = \frac{\log a}{\log b}\), we can rewrite the given function as \(y = \frac{\log x}{\log 3}\). This new function will graph exactly the same as the original, but allows us to work with a base of 10, which is easier for us.

Step 3: Graph the Function

We could now graph the function using a graphing utility. For this step, you need to use a specific software or online tool that allows plotting. It's important to note that the graph will appear the same whether we are graphing \(y=\log _{3} x\) or \(y = \frac{\log x}{\log 3}\).

Key Concepts

Change-of-Base Property

The change-of-base property is a handy mathematical tool that simplifies working with logarithms of different bases. It states that any logarithm can be expressed in terms of base 10 (or any other base) using the formula:

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

This property allows us to convert a logarithm from one base to another, making it easier to compute, especially when no calculator directly supports the original base.
In our problem, we used this property to convert the base 3 logarithm \(y = \log_3 x\) into a base 10 logarithm: \(y = \frac{\log x}{\log 3}\). This transformation simplifies the graphing process as base 10 logarithms are more conventionally supported by calculators and graphing utilities.
Understanding this property is key to tackling logarithms with uncommon bases and helps in various mathematical applications.

Graphing Utility

A graphing utility is a tool, usually in the form of software or an online application, that allows users to visualize mathematical functions on a coordinate plane. These utilities support various functions, including logarithmic functions, helping users to better understand their behavior.
Using a graphing utility is beneficial when you are dealing with complex functions or unfamiliar bases, as in the case of \(y = \log_3 x\). By converting this into a base 10 function \(y = \frac{\log x}{\log 3}\), the utility can easily plot it.

• They let you input the function directly.
• They show you the graph in a clear, visual format.
• They provide additional features like zooming, tracing function points, and more.

When facing questions regarding the graphing of functions, graphing utilities can clarify your understanding. This visualization allows for an intuitive grasp of how the function behaves across different inputs.

Base 10 Logarithms

Base 10 logarithms, commonly known as common logarithms, are widely used because of their straightforward application and their presence on most scientific calculators. The base 10 is notated as \(\log_{10} a\) or simply \(\log a\), assuming base 10 unless otherwise specified.
They simplify calculations, especially in real-world applications such as in scientific, engineering, and financial contexts. By rewriting expressions in terms of base 10, more straightforward computational methods can be employed.
In our conversion using the change-of-base property, we wrote \(y = \frac{\log x}{\log 3}\), enabling easy use of base 10 logarithms for graphing utilities. Base 10 logarithms are not constrained by varying bases, providing a universal approach to handling logarithmic expressions.