Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) Are the expressions equivalent? Explain. Verify your conclusion algebraically. $$y_{1}=\ln (x-2)+\ln (x+2), \quad y_{2}=\ln \left(x^{2}-4\right)$$.

Step-by-Step Solution

Answer

Yes, the expressions are equivalent. The confirmation was both graphically - where the graphs of the two overlapped - and algebraically; with \(y_{1}=\ln (x-2)+\ln (x+2)\) simplified to \(y_{1}=\ln \left(x^{2}-4\right)\), identical to \(y_{2}.\)

Step 1: Graphical Comparison

Using a graphing utility, plot both \(y_{1}=\ln (x-2)+\ln (x+2)\) and \(y_{2}=\ln \left(x^{2}-4\right)\) on the same viewing window. Adjust the window size as necessary to capture a broad range of x-values. If the graphs coincide, then the expressions are equivalent in the graphical sense.

Step 2: Table Creation

Utilize the table feature of the graphing utility to generate corresponding x and y values for both functions. Compare the set of points for each equation. If the sets coincide, it provides further evidence towards the functions' equivalence.

Step 3: Algebraic Comparison

Exploit the properties of logarithms to manipulate either of the expressions to the form of the other. Recall the property: \(\ln(a) + \ln(b) = \ln(ab)\). Applying this property to \(y_{1}\) yields: \(y_{1}=\ln (x-2)+\ln (x+2)=\ln((x-2)(x+2))=\ln (x^{2}-4)\), which is identical to the second expression \(y_{2}\).

Step 4: Verification

Cross-verify by substituting a few x-values into both equations. Should they return the same y-values, it conclusively confirms equivalence of the expressions.

Key Concepts

Graphing Utility

In mathematics, a graphing utility is an invaluable tool for visualizing functions. It comes in handy when comparing expressions like \( y_{1} = \ln (x-2) + \ln (x+2) \) and \( y_{2} = \ln (x^{2}-4) \). A graphing utility allows you to graph both equations within the same viewing window. This lets you visually compare the two functions. By observing if their graphs overlap completely, you can infer that the functions are equivalent in terms of their graphical representation. Additionally, adjusting the window size to encompass a broad spectrum of x-values ensures a comprehensive view of the behavior of each function. This step provides an immediate visual affirmation of equivalence before delving into more detailed numerical or algebraic comparisons.

Logarithmic Properties

The properties of logarithms are fundamental to simplify and manipulate logarithmic expressions. For instance, the property \( \ln(a) + \ln(b) = \ln(ab) \) is crucial for understanding the given problem. Applying this property to \( y_{1} \) yields:

• \( y_{1} = \ln (x-2) + \ln (x+2) \)
• \( = \ln((x-2)(x+2)) \)
• \( = \ln(x^2 - 4) \)

This transformation shows that the expression for \( y_{1} \) simplifies directly to \( y_{2} \), demonstrating their equivalence. This property essentially tells us how logarithms combine when multiplying numbers inside the logarithms, which is useful for verifying whether two expressions that appear different are actually the same.

Algebraic Verification

Algebraic verification is the meticulous process of proving an expressed equivalence by using algebraic manipulation. In this task, after graphically noting the similarity, you confirm it with algebra. Start with simplifying \( y_{1} \) using logarithmic properties as mentioned. Since

• \( y_{1} = \ln (x-2) + \ln (x+2) \equiv \ln (x^{2}-4) \), which simplifies to \( y_{2} \)

Both expressions become identically the same. This exercise not only confirms what the graph showed but also extends understanding of how mathematical identities work. This logical proof reassures that the graphical similarity was not coincidental, but rather a fundamental truth due to the nature of logarithms in algebra.

Graphical Comparison

Graphical comparison supplements algebraic verification by providing a visual confirmation of mathematical relationships. When graphed

• \( y_{1} = \ln (x-2) + \ln (x+2) \) and
• \( y_{2} = \ln (x^{2}-4) \)

they should, under ideal conditions, coincide. By using the table feature in a graphing utility, you can generate specific x and y values for comparison. If each pair of values for the same x matches across both functions, it lends additional evidence to affirm their equivalence. This visual & numerical evidence consolidates understanding and is especially useful when teaching or learning, as it connects abstract algebraic principles to tangible, observable outcomes.