Problem 78 - Algebra A Combined Function

Question

Determine whether each statement is true or false. $$ \left(\log _{3} 6\right) \cdot\left(\log _{3} 4\right)=\log _{3} 24 $$

Step-by-Step Solution

Answer

The statement is false according to logarithm properties.

Step 1: Understand the Given Logarithmic Expression

The exercise involves determining the truth of an equation involving logarithms with the same base 3. The left side is the product of two logarithms with the same base, and the right side is a single logarithm with the same base.

Step 2: Recall the Logarithm Product Rule

The logarithm product rule states that \( \log_b(xy) = \log_b(x) + \log_b(y) \). This rule provides a way to combine the products inside a logarithm into a single logarithm.

Step 3: Apply the Logarithm Product and Power Rule

To determine if the given expression is true, check if multiplying the two logarithms on the left side of the equation can produce the result on the right. No general property of logarithms allows \( (\log_b x)(\log_b y) \) to simplify to \( \log_b xy \). Thus, the expression is incorrect by general logarithm properties.

Step 4: Confirm with a Counter Example

Let's manually calculate both sides for a base 3 log with specific values. Take \( \log_3 6 \approx 1.631 \) and \( \log_3 4 \approx 1.261 \). Then their product is approximately \( 1.631 \times 1.261 \approx 2.057 \). But \( \log_3 24 \approx 2.897 \). Thus, the value check confirms that the equation does not hold true. This counter example further verifies our understanding of logarithm properties.

Key Concepts

Logarithm Product Rule

The logarithm product rule is a handy tool when working with logarithmic expressions. It allows us to transform the logarithm of a product into a sum. In mathematical terms, the rule states that:

• \( \log_b(xy) = \log_b(x) + \log_b(y) \)

To understand how this rule works, consider two numbers, \( a \) and \( b \), under the same logarithmic base. Rather than finding the logarithm of their product directly, we can break it down by calculating the logarithm of each number separately, then adding the results. This not only simplifies calculations but is also essential for solving many logarithmic equations. Using this rule can be vital when dealing with large exponentials or in various algebraic manipulations. Remember, however, this rule only applies to sums, not to products of logarithms.

Base of Logarithm

Every logarithm has a specified base, often denoted as \( b \), that tells you what power a number must be raised to equal another number. For example, in the logarithmic expression \( \log_3 6 \), 3 is the base. The expression answers the question: "To what power must 3 be raised to get 6?"
The base is crucial as it defines how the logarithm behaves. Common logarithmic bases include 10, known as the common logarithm, and \( e \), the natural logarithm. However, the base can be any positive number except 1. In problems, make sure to pay attention to the base provided, as calculations will vary significantly between different bases.
Choosing the right base often depends on the context of the problem or application, whether it's calculating scientific data or financial growth. The consistency of the base throughout an equation is essential to apply logarithmic rules correctly.

Logarithm Properties

When dealing with logarithms, it's important to understand their fundamental properties, which help simplify and solve logarithmic equations. Some key properties include:

• Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• Quotient Rule: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
• Power Rule: \( \log_b(x^n) = n\log_b(x) \)
• Change of Base Formula: \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \) for any positive base \( k \)

These properties allow us to manipulate logarithmic expressions in various ways to simplify or solve equations. For example, the exercise from above attempted to apply these properties to determine if two different forms of logarithmic expressions are equal. Knowing these properties helps avoid common pitfalls, like incorrectly assuming that the product of two logarithms equals the logarithm of a product.