Problem 93 - College Algebra

Question

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(\quad y=2+\log _{3} x, \quad y=\log _{3}(x+2), \quad\) and \(y=-\log _{3} x\) in the same viewing rectangle as \(y=\log _{3} x .\) Then describe the change or changes that need to be made to the graph of \(y=\log _{3} x\) to obtain each of these three graphs.

Step-by-Step Solution

Answer

Adding a constant outside the log function results in a vertical shift of the graph. Adding a constant within the logarithm function results in a horizontal shift. Applying a negative sign to the logarithm function leads to a reflection of the graph over the x-axis. Therefore, to obtain the graphs of \(y=2+\log_{3}(x)\), \(y=\log_{3}(x+2)\), and \(y=-\log_{3}(x)\) from \(y=\log_{3}(x)\), a vertical shift, a horizontal shift, and a reflection over the x-axis are required respectively.

Step 1: Graphing the base function

First, graph the function \(y=\log_{3}(x)\). Note how the function behaves and where its key points are.

Step 2: Graphing manipulated functions and observing changes

Next, graph the functions \(y=2+\log_{3}(x)\), \(y=\log_{3}(x+2)\), and \(y=-\log_{3}(x)\) in the same viewing rectangle as \(y=\log_{3}(x)\). Study the transformations that occur.

Step 3: Understanding transformation principles

The function \(y=2+\log_{3}(x)\) represents a vertical shift of two units upwards from the base function. The function \(y=\log_{3}(x+2)\) means the graph is shifted horizontally to the left by two units. In the case of \(y=-\log_{3}(x)\), the negative sign causes a reflection of the base function over the x-axis.

Step 4: Detailed explanations of transformations

Provide detailed explanations for each transformation. Explain that adding a number outside the logarithmic function results in a vertical shift, adding a number within the logarithm function results in a horizontal shift, and a negative sign leads to a reflection of the graph. The direction of the transformation depends on the sign of the number being added or the negative sign.

Key Concepts

Graphing Logarithms

Logarithmic functions are as crucial to understand as their exponential counterparts, and graphing them helps in visualizing their behavior. Logging a function involves plotting points on a graph where the y-coordinate is the logarithm of the x-coordinate. For instance, when graphing the basic logarithmic function, such as \( y = \log_{3}x \), it's important to know that the graph will have a vertical asymptote at \( x = 0 \) and will pass through the point \( (1,0) \) because \( \log_{3}1 = 0 \). The curve will increase slowly, never touching the y-axis, reflecting the property of logarithms that they are undefined for non-positive numbers.

Moreover, using a graphing utility makes this task easier and more precise, especially when changes are made to the base logarithmic function, as in the exercise. It's important to precisely plot several points on the graph and observe the course and behavior of the function as it moves from left to right, keeping in mind its asymptotic and uninterrupted nature.

Logarithmic Properties

Understanding logarithmic properties is essential when working with these functions, as it aids in transformations and solving equations. The most relevant properties include the product, quotient, and power rules.

• The product rule states that \( \log_b(mn) = \log_b(m) + \log_b(n) \), which allows you to separate the logarithm of a product into the sum of the logarithms.
• The quotient rule is indeed similar, showing that \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \).
• Lastly, the power rule indicates that \( \log_b(m^n) = n \cdot \log_b(m) \), allowing the exponent on a logarithmic argument to be brought out as a multiplier.

These properties are especially useful when simplifying complex logarithmic expressions before graphing. They also come into play when understanding the steps in the process of transformation, as they explain why logarithms behave the way they do and how simple arithmetic modifications alter their graph.

Transformations of Functions

Transformations of logarithmic functions follow the same rules as transformations of other types of functions. There are typically four types of transformations: translations (shifts), reflections, stretches, and compressions.

Translations

Adding or subtracting a value to the function's argument—inside the logarithm—translates the graph horizontally. For instance, \( y = \log_{3}(x+2) \) shifts the graph of \( y = \log_{3}x \) two units to the left. On the contrary, adding or subtracting outside the logarithm, such as \( y=2+\log_{3}x \), translates the graph vertically. This moves the graph two units upwards.

Reflections and Scale Changes

Reflecting a function across one of the axes is achieved by multiplying by -1. For the logarithmic function, putting a negative in front of the function, as in \( y = - \log_{3}x \), reflects it over the x-axis. If we were to multiply inside the argument by -1, the graph would mirror over the y-axis. Additionally, multiplying the entire function by a constant stretches or compresses it vertically, depending on whether the constant is greater than or less than one.

Understanding these transformations is imperative to accurately predict and plot the behavior of a transformed logarithmic function on a graph. These principles allow for a deeper comprehension of the function's geometry and its applications.