Problem 82 - College Algebra

Question

In Exercises \(79-82,\) 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 function \(y = \log_{3}{(x-2)}\) can be graphed using a graphing utility by first applying the change-of-base property to rewrite it as \(y = \frac{log_{10}(x-2)}{log_{10}(3)}\). The graph begins at x=2 and increases slowly as x moves rightward.

Step 1: Understand the Change-of-Base Property

The change-of-base property of logarithms states that for any positive real numbers \(a\), \(b\), and \(c\), where \(a≠1\) and \(b≠1\), the logarithm base \(b\) of \(c\) can be written as the ratio of the logarithm of \(c\) and the logarithm of \(b\), both with the same base\(a\). This can be mathematically represented as \(\log_{b}(c) = \frac{\log_{a}(c)}{\log_{a}(b)}\).

Step 2: Apply the Change-of-Base Formula

For the logarithmic function we have \(y=\log_{3}(x-2)\). Applying the change of base formula, choosing base 10 which is suitable for graphing, this can be rewritten as: \(y = \frac{log_{10}(x - 2)}{log_{10}(3)}\).

Step 3: Graph the function

Input the rearranged formula into the graphing utility. This will typically involve inputting 'y = log(x-2)/log(3)' as the function to be graphed. Once the function is graphed, it can be noted that this is a standard logarithmic graph which has been translated 2 units to the right, due to the presence of the '-2' term inside the function. The graph starts from x=2, moves rightwards and increases slowly, reflecting the characteristics of a typical logarithm function.

Key Concepts

Logarithmic Functions

Logarithmic functions are closely related to exponential functions and serve as their inverse. In simpler terms, if you have an exponential function like \( b^y = x \), the corresponding logarithmic function would be \( y = \log_b(x) \). Logarithms tell us the power to which a base must be raised to obtain a certain number.

• In the example of \( y = \log_3(x-2) \), the logarithm is base 3, meaning it describes the exponent to which 3 must be raised to result in \(x-2\).
• When graphing, it's important to recognize that logarithmic functions have a vertical asymptote, which reflects that they are undefined for non-positive numbers.
• The domain of a logarithmic function with base \( b \) and input \( x-2 \) is \( x > 2 \), because \( (x-2) \) must be positive.

Logarithmic functions tend to grow slowly and never touch the y-axis, representing their continuous increase.

Graphing Utilities

Graphing utilities are tools used to visually represent mathematical functions and explore their behaviors. They can be physical tools, like a graphing calculator, or digital platforms, like graphing software.

• These tools allow you to input a mathematical function and see its graph, providing a clearer understanding of the function's growth and transformations.
• When using a graphing utility for \( y = \log_3(x-2) \), you would input it in the simplified form \( y = \frac{\log_{10}(x-2)}{\log_{10}(3)} \) after applying the change-of-base property.
• This approach helps when the utility only accepts base 10 or natural logarithms.

By observing the graph, you gain insight into characteristics like domain, asymptotes, and potential transformations.

Base of Logarithm

The base of a logarithm is a crucial component that determines the rate of growth within the function. Most mathematical problems use base 10 (common logarithm) or base \( e \) (natural logarithm), but other bases, like 3 in this exercise, are also significant.

• The base indicates how the logarithmic function scales and impacts the shape of the graph.
• In \( y = \log_3(x-2) \), base 3 results in a steeper curve compared to base 10 or \( e \), since fewer iterations are needed for larger outcomes.
• Using the change-of-base property, any base can be converted to another, aiding graphing situations where a specific base is required.

Understanding the base is crucial as it governs the logarithmic function's behavior and its transformation requirements.

Function Transformation

In mathematics, function transformation involves altering a function's position or form. This can include translations, reflections, stretches, and compressions.

• In \( y = \log_3(x-2) \), the \( -2 \) inside the parenthesis represents a horizontal shift.
• This is a transformation where the graph of \( \log_3(x) \) is moved 2 units to the right.
• Other transformations could include changing the base, reflecting across axes, or varying coefficients, which alter the graph's orientation and scale.

Understanding function transformations is key to predicting how adjustments impact the graph of a function, making complex equations much more approachable visually.