Problem 73
Question
Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{k} \frac{p q^{2}}{m}$$
Step-by-Step Solution
Verified Answer
\( \log_k (p) + 2 \cdot \log_k (q) - \log_k (m) \).
1Step 1: Apply the Quotient Rule of Logarithms
The quotient rule of logarithms states that \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \). Applying this rule, the expression becomes: \( \log_{k} (pq^2) - \log_{k} (m) \).
2Step 2: Apply the Product Rule of Logarithms
The product rule of logarithms states that \( \log_b(xy) = \log_b(x) + \log_b(y) \). Applying this to \( \log_k (pq^2) \), we get \( \log_k (p) + \log_k (q^2) \). Thus, the expression is now: \( \log_k (p) + \log_k (q^2) - \log_k (m) \).
3Step 3: Apply the Power Rule of Logarithms
The power rule of logarithms states that \( \log_b(x^n) = n \cdot \log_b(x) \). Applying this to \( \log_k (q^2) \), we get \( 2 \cdot \log_k (q) \). Incorporating this, the expression becomes: \( \log_k (p) + 2 \cdot \log_k (q) - \log_k (m) \).
Key Concepts
Quotient Rule of LogarithmsProduct Rule of LogarithmsPower Rule of Logarithms
Quotient Rule of Logarithms
The quotient rule of logarithms is a fundamental concept that allows us to simplify the logarithm of a fraction. Imagine you have a fraction \( \frac{x}{y} \) and you wish to find its logarithm. The quotient rule tells us that this can be rewritten as the difference of two separate logarithms: \( \log_b\left( \frac{x}{y} \right) = \log_b(x) - \log_b(y) \).
This is extremely useful, especially when dealing with complex expressions where each component of the fraction can be further simplified.
For example, in the expression \( \log _{k} \frac{p q^{2}}{m} \), applying the quotient rule transforms it into \( \log_k (pq^2) - \log_k (m) \).
By doing this, the task of handling the fraction becomes a task of subtracting one logarithm from another, thereby reducing the complexity of the original problem.
Think of the quotient rule as a powerful tool in breaking down bigger logarithmic problems into smaller, more manageable parts.
This is extremely useful, especially when dealing with complex expressions where each component of the fraction can be further simplified.
For example, in the expression \( \log _{k} \frac{p q^{2}}{m} \), applying the quotient rule transforms it into \( \log_k (pq^2) - \log_k (m) \).
By doing this, the task of handling the fraction becomes a task of subtracting one logarithm from another, thereby reducing the complexity of the original problem.
Think of the quotient rule as a powerful tool in breaking down bigger logarithmic problems into smaller, more manageable parts.
Product Rule of Logarithms
The product rule of logarithms is another key tool in simplifying logarithmic expressions. It is used when you have the logarithm of two numbers multiplied together, such as \( x \cdot y \). The product rule states that \( \log_b(xy) = \log_b(x) + \log_b(y) \).
This rule helps us deconstruct products inside a logarithm into a simple sum of individual logarithms, making it easier to work with each component separately.
Going back to our exercise, after applying the quotient rule, we have \( \log_k (pq^2) - \log_k (m) \).
Applying the product rule to \( \log_k (pq^2) \) gives us \( \log_k (p) + \log_k (q^2) \). This results in the intermediate expression: \( \log_k (p) + \log_k (q^2) - \log_k (m) \).
Breaking apart the product in this way lays the groundwork for applying other rules, such as the power rule, and helps simplify complex logarithmic calculations step by step.
This rule helps us deconstruct products inside a logarithm into a simple sum of individual logarithms, making it easier to work with each component separately.
Going back to our exercise, after applying the quotient rule, we have \( \log_k (pq^2) - \log_k (m) \).
Applying the product rule to \( \log_k (pq^2) \) gives us \( \log_k (p) + \log_k (q^2) \). This results in the intermediate expression: \( \log_k (p) + \log_k (q^2) - \log_k (m) \).
Breaking apart the product in this way lays the groundwork for applying other rules, such as the power rule, and helps simplify complex logarithmic calculations step by step.
Power Rule of Logarithms
The power rule of logarithms shines when you have an exponent within a logarithm. According to this rule, if you have \( x^n \), it can be expressed as \( n \cdot \log_b(x) \). This means you take the exponent and place it in front of the logarithm.
This is particularly useful because it reduces the problem of dealing with powers into a simpler multiplication problem.
In the exercise problem, after applying the quotient and product rules, we arrive at \( \log_k (p) + \log_k (q^2) - \log_k (m) \).
Now by applying the power rule to \( \log_k (q^2) \), it becomes \( 2 \cdot \log_k(q) \).
Thus, the final expression is simplified to \( \log_k (p) + 2 \cdot \log_k (q) - \log_k (m) \).
This step-by-step log transformation using power rules is crucial for simplifying complex expression efficiently.
This is particularly useful because it reduces the problem of dealing with powers into a simpler multiplication problem.
In the exercise problem, after applying the quotient and product rules, we arrive at \( \log_k (p) + \log_k (q^2) - \log_k (m) \).
Now by applying the power rule to \( \log_k (q^2) \), it becomes \( 2 \cdot \log_k(q) \).
Thus, the final expression is simplified to \( \log_k (p) + 2 \cdot \log_k (q) - \log_k (m) \).
This step-by-step log transformation using power rules is crucial for simplifying complex expression efficiently.
Other exercises in this chapter
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