Question

Match the expression with the logarithm that has the same value. Justify your answer. A. \(\log _3 64\) B. \(\log _3 3\) C. \(\log _3 12\) D. \(\log _3 36\) $$\log _3 6-\log _3 2$$

Short answer

\(\log _3 6 - \log _3 2\) matches with B. \(\log _3 3\).

Step-by-step solution

step 1: Apply Logarithm Properties

Use one of the properties of logarithms, which states that \(\log b(a) - \log b(c)\) equals \(\log b(\frac{a}{c})\). Apply it to the expression \(\log _3 6-\log _3 2\), and it becomes \(\log _3 (\frac{6}{2})\).

Simplify the Expression

Simplify the fraction \(\frac{6}{2}\), and the expression becomes \(\log _3 3\).

Match with the Correct Expression

Now, match the simplified expression \(\log _3 3\) with the given expressions. Upon inspection, that is option B.

Key concepts

Logarithm Properties

Logarithm properties are powerful tools that can make complex expressions much simpler. Understanding these properties can help us handle logarithmic expressions with ease. The key property used in this exercise is the logarithmic subtraction rule. This rule states that the difference between two logarithms with the same base, such as \(\log_b(a) - \log_b(c)\), is equal to \(\log_b(\frac{a}{c})\). This means that when you subtract one logarithm from another with the same base, you can combine them into a single logarithm related to the division of their arguments.

• Product Property: \(\log_b(mn) = \log_b(m) + \log_b(n)\)
• Quotient Property: \(\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)\)
• Power Property: \(\log_b(m^n) = n \cdot \log_b(m)\)

Using these properties, we can often make logarithmic calculations without a calculator, simplifying the problem mentally.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is about making them more concise while maintaining their original value. In our exercise, we were given the task of simplifying \(\log_3 6-\log_3 2\). By using the logarithmic quotient property mentioned earlier, we simplify it to \(\log_3 \frac{6}{2}\). Once simplified further, \(\frac{6}{2}\) equals \(3\), hence the expression becomes \(\log_3 3\).
Understanding this step is crucial because:

• It reduces the complexity of the expression.
• Makes it easier to compare with other expressions.

The act of simplifying is part understanding the rules and part arithmetic—both working together to make the expression simpler and more manageable.

Matching Logarithmic Expressions

Matching logarithmic expressions involves identifying expressions that are equivalent. This process is like solving a puzzle, aligning pieces that fit with each other. In the exercise, after simplifying \(\log_3 6-\log_3 2\) to get \(\log_3 3\), we compared it with the options provided:

• A. \(\log_3 64\)
• B. \(\log_3 3\)
• C. \(\log_3 12\)
• D. \(\log_3 36\)

By simple inspection, \(\log_3 3\) matches exactly with option B. This step requires understanding both the properties of logarithms and basic number matching techniques.
When you match correctly, you confirm your solution maintains consistent value throughout transformations, and everything aligns perfectly.