Problem 86 - Precalculus

Question

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

Step-by-Step Solution

Answer

Yes, they are the same graph because both simplify to \( y = \log \left( \frac{x}{2} \right) \).

Step 1: Understand the Functions

We have two functions: \( y = \log \left( \frac{x}{2} \right) \) and \( y = \log x - \log 2 \). We need to determine if these two expressions represent the same function by analyzing and plotting them using a graphing calculator.

Step 2: Simplify the Second Function

The function \( y = \log x - \log 2 \) can be simplified using the logarithmic identity: \( \log a - \log b = \log \left( \frac{a}{b} \right) \). Applying this identity, we get \( y = \log \left( \frac{x}{2} \right) \).

Step 3: Compare the Functions

Both functions simplify to \( y = \log \left( \frac{x}{2} \right) \). This algebraic simplification shows that both originally different-looking expressions define the same function.

Step 4: Graph the Functions

To visualize the functions, graph both \( y = \log \left( \frac{x}{2} \right) \) and \( y = \log x - \log 2 \) using a graphing calculator. Adjust the viewing window to ensure you can see the behavior of both graphs clearly, especially in the domain where \( x > 0 \).

Step 5: Analyze the Graphs

Observe the graph on the calculator screen. Both equations, when graphed, should produce the same line, reaffirming our algebraic simplification. If both lines overlap completely on the graph, it confirms they are the same function.

Key Concepts

Logarithmic Functions

Logarithmic functions are a key concept in algebra and are used to express the inverse operations of exponentiation. Consider the two given functions:

• First function: \( y = \log \left( \frac{x}{2} \right) \)
• Second function: \( y = \log x - \log 2 \)

These functions utilize the logarithm, which indicates the power to which a number must be raised to obtain another number. When working with logarithmic functions, graphing calculators are invaluable as they allow us to visualize these equations fully.

When plotted on a graphing calculator, these logarithmic expressions can reveal similarities, such as overlapping lines, indicating that they convey the same relationship. This is vital for understanding how seemingly different expressions can actually describe the same function in mathematical terms.

Algebraic Simplification

In mathematics, algebraic simplification is a process used to rewrite complex expressions in a simpler form without changing their value. We started with two expressions: \( y = \log x - \log 2 \) and \( y = \log \left( \frac{x}{2} \right) \). Using algebraic simplification, we can transform the second expression using a logarithmic identity:

• Simplify \( \log x - \log 2 \) using the identity \( \log a - \log b = \log \left( \frac{a}{b} \right) \).
• This gives us \( y = \log \left( \frac{x}{2} \right) \).

Through this simplification, we discovered that both original logarithmic functions actually represent the same mathematical function. The process of simplifying such expressions is crucial in algebra, as it helps clarify and reduce complexity, allowing for a clearer understanding and visualization.

Logarithmic Identity

Logarithmic identities are powerful tools in algebra that simplify logarithmic expressions, facilitating easier computation and analysis. The identity used in our example is:

• \( \log a - \log b = \log \left( \frac{a}{b} \right) \)
• This means that when you subtract logarithms of two numbers, you can express it as the logarithm of a division of those numbers.

Using this logarithmic identity, we confirm that \( y = \log x - \log 2 \) simplifies to \( y = \log \left( \frac{x}{2} \right) \).

Identities like these are essential, as they help simplify complex problems and reveal underlying patterns. For students and mathematicians alike, understanding and applying logarithmic identities is key to mastering functions and their transformations. In this instance, employing such an identity proves that the two forms of our function are, in fact, identical.