Problem 50 - Algebra 2

Question

CHALLENGE Tell whether each statement is true or false. If true, show that it is true. If false, give a counterexample. For all positive numbers \(m, n, x,\) and \(b,\) where \(b \neq 1, n \log _{b} x+m \log _{b} x=\) \((n+m) \log _{b} x\)

Step-by-Step Solution

Answer

True, the statement is always true by the distributive property of logarithms.

Step 1: Understand the Statement

The statement claims that for any positive numbers \(m, n, x\), and \(b\) (where \(b eq 1\)), the equation \(n \log_{b} x + m \log_{b} x = (n+m) \log_{b} x\) holds true. We need to determine if this equation is always true or if there's a situation where it doesn't hold, which would make the statement false.

Step 2: Simplify the Left Side of the Equation

On the left side of the equation, we have \(n \log_{b} x + m \log_{b} x\). We can factor out \(\log_{b} x\) since it is a common factor: \[ n \log_{b} x + m \log_{b} x = (n + m) \log_{b} x \]

Step 3: Compare Both Sides of the Equation

After factoring out \(\log_{b} x\) from the left side, both sides of the equation are now \( (n+m) \log_{b} x\). This means the left side is equal to the right side of the equation based on the distributive property.

Step 4: Conclusion

The original statement is observed to be true as shown by simplifying the expression and connecting both sides of the equal sign. Since there is no contradiction or example where it does not hold, we affirm the statement is generally true.

Key Concepts

Distributive Property

The distributive property is a fundamental rule in algebra that allows you to multiply a single term across terms within parentheses. It essentially says that a term can be "distributed" over addition or subtraction inside a set of parentheses.
In this exercise, the distributive property is showcased through logarithmic expressions:

• Given the terms: \( n \log_{b} x \) and \( m \log_{b} x \).
• You can factor out the common term, \( \log_{b} x \), resulting in \((n + m) \log_{b} x\).

This simplification relies on recognizing the common factor and distributing it over the sum of \( n \) and \( m \).
Understanding the distributive property with logarithms helps in simplifying complex expressions with ease.

Positive Numbers

Positive numbers are integers or real numbers that are greater than zero. They play a critical role in many mathematical concepts, including the ones involved in this exercise.
In the context of logarithms:

• All the variables \(m, n, x\), and base \(b\) must be positive (with \(b eq 1\)).
• This ensures all logarithmic expressions are defined and valid.

It's essential to use positive numbers to avoid undefined logarithmic calculations or unexpected results.
Their positivity is crucial for maintaining the integrity of expressions throughout the equation manipulation.

Counterexample

A counterexample is a specific case or example that contradicts a general statement, proving it false.
In mathematical terms, if you claim a general rule, one effective way to test it is to look for a counterexample.

• For this exercise, if we had found a set of positive numbers \( m, n, x\) and \(b\) that prove \(n \log_{b} x + m \log_{b} x eq (n+m) \log_{b} x\), it would constitute a counterexample.
• However, the step-by-step solution finds no contradiction, affirming the statement is true.

Attempting to find a counterexample is a powerful tool in mathematics, ensuring claims hold under all given conditions.

Simplification of Expressions

Simplifying expressions is a process of rewriting them in a more concise or easily manageable form.
In this exercise, simplification plays a key role in validating the equation provided.

• The left-hand side expression \(n \log_{b} x + m \log_{b} x\) could appear complex initially.
• However, by factoring out the \(\log_{b} x\), it simplifies to \((n + m) \log_{b} x\).

This process showcases how simplifying helps in identifying equalities or verifying the truth of expressions.
Simplification is an invaluable skill in algebra, reducing errors and providing a clearer path to the answer.