Problem 82 - Algebra and Trigonometry

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

The graph of the function \(y=\log _{3}(x-2)\) after applying the change-of-base property trends towards negative infinity as x approaches 2 from the right, and it increases for x values greater than 2, which represents the typical behavior of a logarithmic function.

Step 1: Apply the change-of-base property

The first step is applying the change-of-base property to the given function. Using the formula \(\log_c(a) = \frac{\log_b(a)}{\log_b(c)}\), the function \( y=\log _{3}(x-2) \) is rewritten as \( y = \frac{\log_{10}(x-2)}{\log_{10}(3)} \). Now, it is ready to be graphed using a graphing utility. The base 10 is used because most graphing utilities have built-in functions for the common (base 10) logarithm.

Step 2: Graph the function

Enter the transformed function \(\frac{\log_{10}(x-2)}{\log_{10}(3)} \) into a graphing utility. The range of the function should be that \( x>2 \) because the inside of the logarithm \( x-2 \) has to be greater than zero.

Step 3: Interpret the graph

Observing the graph gives an understanding of the function's behavior. As x approaches 2 from the right, the function tends towards negative infinity. This is due to the logarithmic function approaching negative infinity as its argument approaches zero. As x increases beyond 2, the function value increases, which displays the characteristic shape of a logarithmic graph.

Key Concepts

Change-of-Base Property

The change-of-base property is a handy tool that allows us to compute logarithms with bases that might not be directly supported by calculators or graphing utilities. This property states that for a logarithm with an arbitrary base, say base \( c \), it can be translated into a fraction of logarithms with a different base \( b \). The formula used is

• \( \log_c(a) = \frac{\log_b(a)}{\log_b(c)} \),

where \( a \) is the argument of the logarithm.
This is particularly useful because most calculators and graphing tools primarily support the natural logarithm (base \( e \)) and the common logarithm (base 10).
So, when working with a function like \( y = \log_3(x-2) \), we can rewrite it using base 10 logarithms as \( y = \frac{\log_{10}(x-2)}{\log_{10}(3)} \).
This makes the function compatible with graphing utilities that often have built-in functionality for base 10 calculations. Understanding and applying this property simplifies the graphing process, especially when dealing with less common logarithmic bases.

Graphing Utilities

Graphing utilities are powerful tools used to visualize mathematical functions. They help us create precise and clear graphs based on inputted equations.
These utilities are particularly useful in understanding complex functions like logarithmic functions which may not be intuitive to plot manually.
To graph a function, you can input a modified equation, such as \( y = \frac{\log_{10}(x-2)}{\log_{10}(3)} \), into the utility to see a real-time plot of the function.

• Always remember: for logarithmic functions, the domain must respect any restrictions for the logarithm to be defined. For instance, in the function \( y = \log_3(x-2) \), make sure \( x-2 > 0 \), which implies \( x > 2 \).

This means you should set the graph to only show values where \( x \) is greater than 2, ensuring an accurate representation.
By using graphing utilities, we utilize technology to enhance our understanding of mathematical behaviors and relationships.

Logarithmic Graph Behavior

Logarithmic graphs have distinct characteristics that differentiate them from other types of functions. When graphing a function like \( y = \log_3(x-2) \), certain behavior patterns can be expected.

• As \( x \) approaches 2 from the right (just above 2), the function tends toward negative infinity, because the expression inside the logarithm, \( x-2 \), moves closer to zero.
• As \( x \) increases further beyond 2, the function value begins to increase gradually. This is indicative of the slow growth rate of logarithmic functions.

Logarithmic graphs start by decreasing sharply and then rise quietly, displaying what is often called the 'logarithmic scale'.
This behavior is essential when analyzing data, especially in fields that deal with exponential growth, where log scales make large data sets easier to manage visually.
When examining such graphs, understanding these characteristics helps in predicting and interpreting the changes in the function as \( x \) varies.