Problem 93 - College Algebra with Modeling and Visualization

Question

A student insists that \(\log (x+y)\) and \(\log x+\log y\) are equal. How could you convince the student otherwise?

Step-by-Step Solution

Answer

\( \log(x+y) \neq \log x + \log y \); they are not equal due to the properties of logs.

Step 1: Clarify the Properties of Logarithms

Logarithms have several important properties that help simplify expressions. One key property is that \( \log(ab) = \log a + \log b \), not \( \log(a+b) \). It's important to understand that \( \log(a) + \log(b) \) equates to \( \log(ab) \), meaning the arguments of the logarithms are multiplied, not added.

Step 2: Present a Counterexample

To show that \( \log(x+y) eq \log x + \log y \), consider specific values. Let's use \( x=2 \) and \( y=3 \). Calculate both sides separately: \( \log(2+3) = \log(5) \), and \( \log 2 + \log 3 = \log(6) \). \( \log(5) \) and \( \log(6) \) are not equal.

Step 3: Provide a Numerical Calculation

Using a calculator, compute the values. \( \log(5) \) approximately equals 0.6990 (base 10), and \( \log(6) \) approximately equals 0.7782. It confirms that 0.6990 is not equal to 0.7782, thereby proving \( \log(x+y) eq \log x + \log y \).

Step 4: Reinforce the Understanding

Reiterate that the correct application of logarithms involves the multiplication of arguments when two logs are added, as shown by the property \( \log(a) + \log(b) = \log(ab) \). Adding the arguments directly as in \( \log(x+y) \) doesn't apply this property correctly.

Key Concepts

Counterexample

When dealing with mathematical expressions, a counterexample is a powerful tool to disprove a statement or belief. In our scenario, the student mistakenly thinks that \( \log(x+y) \) equals \( \log x + \log y \). To correct this, we want to provide a counterexample, showing that their belief does not hold true for all values.

To create a counterexample:

• Start by choosing specific numbers for \(x\) and \(y\). In this case, we use \(x = 2\) and \(y = 3\).
• Calculate \( \log(x+y) \), which becomes \( \log(5) \).
• Next, compute \( \log x + \log y \), which results in \( \log(2) + \log(3) = \log(6) \).

After performing the calculations, it is clear that \( \log(5) \) and \( \log(6) \) are not the same, thus effectively disproving the initial statement through a counterexample.
This method serves as a direct indication that the original assumption was mistaken.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a statement. To reinforce why \( \log(x+y) eq \log x + \log y \), we rely on the properties of logarithms and demonstrate it clearly.

Logarithms have specific rules, such as:

• \( \log(ab) = \log a + \log b \)
• \( \log(a+b) \) is not part of this property

Using these properties, we prove that the combination of logs represents multiplication of their arguments, not addition.
Therefore, for \( \log a + \log b \) to accurately add up, it must convert to \( \log(ab) \), meaning the result of multiplying \(a\) and \(b\).

By understanding these fundamental concepts and examining specific values, students can grasp why the statement \( \log(x+y) = \log x + \log y \) fails under mathematical scrutiny.

Educational Strategies

Teaching complex concepts like logarithmic properties can be challenging. Here are some educational strategies to help convey these ideas clearly:

• Visual Aids: Use graphs and charts to illustrate the properties of logarithms visually. This helps in comprehending the nature of logarithmic functions.



• Interactive Sessions: Encourage students to manipulate logarithmic expressions themselves or use a calculator to test their results with different values. Engagement solidifies understanding.



• Real-World Applications: Show how logarithms apply to real-life scenarios. Relating math to practical applications can make abstract concepts more tangible.

By applying these methods, students are more likely to engage with the material and understand the underlying principles, improving their retention and application of logarithmic properties.