Problem 83 - Precalculus

Question

Use a graphing calculator to plot \(y=\ln (2 x)\) and \(y=\ln 2+\ln x .\) Are they the same graph?

Step-by-Step Solution

Answer

Yes, they are the same graph.

Step 1: Understanding the Functions

The functions given are \( y = \ln(2x) \) and \( y = \ln 2 + \ln x \). We need to determine if these represent the same mathematical relationship.

Step 2: Applying Logarithm Properties

Use the logarithm property \( \ln(ab) = \ln a + \ln b \) to transform \( \ln(2x) \). According to this property, \( \ln(2x) = \ln 2 + \ln x \).

Step 3: Simplifying the Functions

Since \( \ln(2x) = \ln 2 + \ln x \) algebraically simplifies to \( y = \ln 2 + \ln x \), these two expressions are mathematically equivalent under the property of logs.

Step 4: Confirming with a Graphing Calculator

Plot both functions \( y = \ln(2x) \) and \( y = \ln 2 + \ln x \) on a graphing calculator. Check if the graphs overlap completely to confirm their equivalence.

Step 5: Conclusion

As both functions simplify to the same expression and their graphs overlap exactly, they represent the same graph.

Key Concepts

Properties of Logarithms

Logarithmic functions have specific properties that simplify complex expressions. These properties make calculations involving logarithms much easier. Let's focus on a key property used in our exercise: the product property.
The product property of logarithms states that \( \ln(ab) = \ln a + \ln b \). This means, when you have a logarithm of a product, you can break it into a sum of logarithms.
In the exercise, we used this property to show that \( \ln(2x) = \ln 2 + \ln x \). By applying this property, the original expression is simplified, highlighting the mathematical equivalence of the two expressions.

• This property is essential for understanding how seemingly different looking expressions can actually represent the same function.
• It helps tremendously in simplifying and solving logarithmic equations.

This foundational property is widely used in higher mathematics and practical applications, allowing for more straightforward manipulation of logarithmic expressions.

Graphing Calculators

Graphing calculators are crucial tools for visualizing mathematical concepts. They allow us to plot equations and observe their graphical behavior. In the context of logarithmic functions, they are especially valuable.
By plotting the functions \( y = \ln(2x) \) and \( y = \ln 2 + \ln x \), a graphing calculator can visually confirm their equivalence.
When you input these equations into a graphing calculator:

• You'll see both graphs appear on the screen.
• If the graphs overlap completely, it confirms that the functions are equivalent.

Graphing calculators provide a straightforward method for students to verify solutions and develop a deeper understanding of mathematical relationships. They bridge the gap between abstract mathematical concepts and tangible visual evidence.

Mathematical Equivalence

Mathematical equivalence is a fundamental concept indicating that two expressions represent the same value. In the exercise, determining the equivalence of \( \ln(2x) \) and \( \ln 2 + \ln x \) illustrates this principle.
Using the property of logarithms, we showed their algebraic equivalence, which is crucial for solving equations and validating solutions.
Equivalence means:

• The expressions hold true across all values within a given domain.
• They can be used interchangeably, meaning solving one gives the same result as solving the other.

In practice, confirming equivalence allows for flexible problem-solving approaches and builds a deeper understanding of how different algebraic transformations relate. Understanding mathematical equivalence nurtures skills crucial for higher-level mathematics.