Problem 46
Question
Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\log _{2} \frac{a b}{4}\)
Step-by-Step Solution
Verified Answer
\(\log_2a + \log_2b - 2\)
1Step 1: Express the Logarithm Using the Quotient Rule
The quotient rule of logarithms states that \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\). Here, apply this rule to express \(\log_2 \frac{ab}{4}\) as \(\log_2(ab) - \log_2(4)\).
2Step 2: Use the Product Rule for Logarithms
The product rule for logarithms tells us that \(\log_b(MN) = \log_b M + \log_b N\). Use this rule to express \(\log_2(ab)\) as \(\log_2a + \log_2b\).
3Step 3: Simplify the Expression
Combine the results from Steps 1 and 2 to rewrite the expression as a sum and difference: \(\log_2a + \log_2b - \log_2(4)\). Since \(\log_2(4) = 2\), further simplify to get \(\log_2a + \log_2b - 2\).
Key Concepts
Quotient Rule of LogarithmsProduct Rule of LogarithmsSimplifying Logarithms
Quotient Rule of Logarithms
When we encounter the logarithm of a fraction, the quotient rule comes in handy. This rule allows us to separate the logarithm of a quotient, which is a division, into a difference of two logarithms. For instance, if we have the expression \( \log_b \left( \frac{M}{N} \right) \), the quotient rule tells us that this can be rewritten as \( \log_b M - \log_b N \). This transformation lets us handle the terms individually.
In our exercise, we are dealing with \( \log_2 \left( \frac{ab}{4} \right) \). Applying the quotient rule, we express it as \( \log_2(ab) - \log_2(4) \). This breakdown is critical because it simplifies our work later on by isolating each part of the logarithmic expression.
Using the quotient rule is like breaking a big task into small, manageable chunks. Once separated, each term can be worked on individually allowing for further simplification.
In our exercise, we are dealing with \( \log_2 \left( \frac{ab}{4} \right) \). Applying the quotient rule, we express it as \( \log_2(ab) - \log_2(4) \). This breakdown is critical because it simplifies our work later on by isolating each part of the logarithmic expression.
Using the quotient rule is like breaking a big task into small, manageable chunks. Once separated, each term can be worked on individually allowing for further simplification.
Product Rule of Logarithms
The product rule of logarithms is another powerful tool. It transforms a logarithm of a product into a sum of logarithms. According to the product rule: \( \log_b (MN) = \log_b M + \log_b N \). This rule brings clarity when dealing with products inside logarithms.
In the given exercise, the term \( \log_2(ab) \) can be effectively split using the product rule into \( \log_2a + \log_2b \). This expansion makes each variable independent, easing further mathematical manipulation.
Understanding the product rule can be particularly helpful because it allows you to see how each component of a product contributes to the overall value of the logarithm. It facilitates simplification and manipulation, especially when dealing with complex logarithmic expressions.
In the given exercise, the term \( \log_2(ab) \) can be effectively split using the product rule into \( \log_2a + \log_2b \). This expansion makes each variable independent, easing further mathematical manipulation.
Understanding the product rule can be particularly helpful because it allows you to see how each component of a product contributes to the overall value of the logarithm. It facilitates simplification and manipulation, especially when dealing with complex logarithmic expressions.
Simplifying Logarithms
Simplifying logarithms involves reducing expressions to their most basic form. After applying both the quotient and product rules, you're often left with individual logarithmic terms that can be simplified further.
In our solution, after using these rules, we reach the expression \( \log_2a + \log_2b - \log_2(4) \). Recognizing simple numeric logarithms is key. Here, \( \log_2(4) = 2 \) because \(2^2 = 4\). This allows us to replace \( \log_2(4) \) with 2, further simplifying our expression to \( \log_2a + \log_2b - 2 \).
Simplification is about recognizing such straightforward calculations and reducing expressions without losing the essence. It's like cleaning up a sentence by eliminating unnecessary words while preserving the meaning. The goal is to get to a simpler, more concise expression that is easier to understand and solve.
In our solution, after using these rules, we reach the expression \( \log_2a + \log_2b - \log_2(4) \). Recognizing simple numeric logarithms is key. Here, \( \log_2(4) = 2 \) because \(2^2 = 4\). This allows us to replace \( \log_2(4) \) with 2, further simplifying our expression to \( \log_2a + \log_2b - 2 \).
Simplification is about recognizing such straightforward calculations and reducing expressions without losing the essence. It's like cleaning up a sentence by eliminating unnecessary words while preserving the meaning. The goal is to get to a simpler, more concise expression that is easier to understand and solve.
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Problem 46
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