Problem 85 - Precalculus

Question

Use a graphing calculator to plot \(y=\frac{\log x}{\log 2}\) and \(y=\log x-\log 2 .\) Are they the same graph?

Step-by-Step Solution

Answer

No, they are not the same graph.

Step 1: Understand Each Function

The functions you need to plot are \( y = \frac{\log x}{\log 2} \) and \( y = \log x - \log 2 \). Both involve logarithms but are expressed differently. The base for these logarithms is 10 (common logarithm), unless otherwise specified.

Step 2: Plot the First Function

Using your graphing calculator, plot the first function: \( y = \frac{\log x}{\log 2} \). This is effectively the change of base formula, converting a log base 2 into a common logarithm, shown as \( y = \log_2 x \).

Step 3: Plot the Second Function

Now plot the second function: \( y = \log x - \log 2 \). Recall the logarithm property: \( \log a - \log b = \log \frac{a}{b} \). This function can be rewritten as \( y = \log \left( \frac{x}{2} \right) \).

Step 4: Analyze the Graphs

Examine both plots on the calculator. The first equation \( y = \frac{\log x}{\log 2} \) simplifies to \( y = \log_2 x \), which represents the log base 2 of x. The second equation \( y = \log x - \log 2 \) simplifies to \( y = \log \left( \frac{x}{2} \right) \). These equations describe different transformations: the first is a change of base, and the second is a translation.

Step 5: Compare the Graphs

Since \( y = \frac{\log x}{\log 2} = \log_2 x \) and \( y = \log \left( \frac{x}{2} \right) \) are derived differently, the plotted graphs are not the same. The first graph represents a base-change logarithm, and the second graph is a scaled translate of the common log graph.

Key Concepts

Change of Base Formula

The change of base formula is crucial in understanding logarithmic expressions. It helps to convert logarithms from one base to another, which can simplify complex calculations. For any logarithm with base 'a', the formula is given by:

• \( \log_b x = \frac{\log_a x}{\log_a b} \)

This formula is especially helpful when dealing with logarithms that aren't in base 10 or base \( e \). For instance, to convert \( \log_2 x \) to a common logarithm (base 10), you can express it as \( \frac{\log_{10} x}{\log_{10} 2} \).
This transformation allows us to use standard calculators or calculators that do not support the input of different bases directly. In the exercise, the function \( y = \frac{\log x}{\log 2} \) is derived using this change of base formula, which is equivalent to \( \log_2 x \), thereby plotting the logarithm with respect to base 2 using base 10 logarithms.

Logarithmic Transformations

Logarithmic transformations are rules that help to manipulate and simplify logarithmic expressions. These transformations stem from fundamental properties of logarithms:

• \( \log a - \log b = \log \left( \frac{a}{b} \right) \)
• \( \log a + \log b = \log (ab) \)

In the exercise, we use the property \( \log x - \log 2 = \log \left( \frac{x}{2} \right) \) to transform the function \( y = \log x - \log 2 \) into a simpler form.
This transformation helps to visualize how combining logarithms leads to a single log with a quotient inside. The expression moves horizontally on the graph, translating the typical logarithmic graph by dividing x by 2. Understanding these transformations is essential because they illuminate how logarithm scales and translates impact the graph's shape.

Graphing Calculator Usage

Using a graphing calculator is a practical skill for visualizing mathematical functions such as logarithms. To plot complex functions, understand how to input equations accurately. You typically need to:

• Switch to the function graphing mode.
• Enter the equation, using appropriate syntax for logarithms. For a common logarithm, you might input \( \log(x) \).
• Adjust window settings to ensure both axes are scaled to show key features of the graph clearly.

For this particular exercise, plotting \( y = \frac{\log x}{\log 2} \) and \( y = \log(x) - \log(2) \) helps to visualize how different mathematical transformations influence the appearance and behavior of graphs.
By observing the graph, you can see that the first equation represents a change of base, while the second depicts a translated graph of the log base 10 function. Such distinctions illustrate the strength of visualization in understanding functional transformations, affirming the usefulness of graphing calculators in studying mathematical concepts.