Problem 66
Question
Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{4} \frac{6}{7}$$
Step-by-Step Solution
Verified Answer
\( \log_4 \frac{6}{7} = \log_4 6 - \log_4 7 \)
1Step 1: Apply the Quotient Rule for Logarithms
According to the Quotient Rule for logarithms, the logarithm of a quotient is equal to the difference of the logarithms. Therefore, \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \). We can apply this rule to \( \log_4 \frac{6}{7} \).
2Step 2: Rewrite the Original Expression
Using the Quotient Rule, rewrite \( \log_4 \frac{6}{7} \) as \( \log_4 6 - \log_4 7 \).
Key Concepts
Properties of LogarithmsQuotient RuleMathematical ExpressionsCollege Algebra
Properties of Logarithms
Understanding the properties of logarithms is essential for simplifying logarithmic expressions and solving logarithmic equations. These properties include basic rules like the product rule, the quotient rule, and the power rule, among others. They help break down complex expressions into simpler parts, making calculations easier and more intuitive.
For example:
These rules help in rewriting logarithmic expressions into simpler forms, making it easier to evaluate or manipulate them in various contexts. Understanding these properties deeply will allow you to solve problems efficiently.
For example:
- The product rule states that the logarithm of a product is the sum of the logarithms: \( \log_b (xy) = \log_b x + \log_b y \).
- The quotient rule states that the logarithm of a quotient is the difference of the logarithms: \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \).
- The power rule states that the logarithm of a power is the exponent times the logarithm of the base: \( \log_b (x^n) = n \log_b x \).
These rules help in rewriting logarithmic expressions into simpler forms, making it easier to evaluate or manipulate them in various contexts. Understanding these properties deeply will allow you to solve problems efficiently.
Quotient Rule
In the realm of logarithms, the quotient rule is a powerful tool. It allows you to break down a single logarithm of a division into two separate logarithms, which is incredibly useful for further simplification. The rule states:
In our example, applying the quotient rule to \( \log_4 \frac{6}{7} \) simplifies it to \( \log_4 6 - \log_4 7 \). This form is often easier to handle, particularly in algebraic manipulations and calculations.
- \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \)
In our example, applying the quotient rule to \( \log_4 \frac{6}{7} \) simplifies it to \( \log_4 6 - \log_4 7 \). This form is often easier to handle, particularly in algebraic manipulations and calculations.
Mathematical Expressions
Mathematical expressions are combinations of numbers, variables, and operations that create an entity representing a particular value or relationship. In the context of logarithms, mathematical expressions often involve one or more of the properties of logarithms to reach a simpler form or solve for unknown values.
When dealing with logarithms, expressions can become complex due to the many rules and transformations applied. However, rewriting them using properties, like the quotient rule, aids greatly.
For example, the expression \( \log_4 \frac{6}{7} \) can initially seem complex, but employing the quotient rule breaks it into simpler parts: \( \log_4 6 - \log_4 7 \). This highlights the importance of understanding and utilizing the properties of logarithms to simplify expressions and solve problems effectively.
When dealing with logarithms, expressions can become complex due to the many rules and transformations applied. However, rewriting them using properties, like the quotient rule, aids greatly.
For example, the expression \( \log_4 \frac{6}{7} \) can initially seem complex, but employing the quotient rule breaks it into simpler parts: \( \log_4 6 - \log_4 7 \). This highlights the importance of understanding and utilizing the properties of logarithms to simplify expressions and solve problems effectively.
College Algebra
College algebra often introduces students to more advanced mathematical concepts that build on their prior knowledge. Logarithms are a significant part of this subject, as they are foundational to topics in calculus and other higher-level mathematics.
Courses in college algebra delve deep into properties of logarithms and how they extend to rational and irrational numbers, their transformations, and their applications in various fields. Students are expected to learn how to manipulate logarithmic expressions, solve logarithmic equations, and apply these skills in real-world contexts.
In college algebra, problems like rewriting \( \log_4 \frac{6}{7} \) using the quotient rule prepare students for more complex equations. Mastery of these concepts not only boosts mathematical intuition but also enhances analytical skills, which are invaluable in academic and professional endeavors.
Courses in college algebra delve deep into properties of logarithms and how they extend to rational and irrational numbers, their transformations, and their applications in various fields. Students are expected to learn how to manipulate logarithmic expressions, solve logarithmic equations, and apply these skills in real-world contexts.
In college algebra, problems like rewriting \( \log_4 \frac{6}{7} \) using the quotient rule prepare students for more complex equations. Mastery of these concepts not only boosts mathematical intuition but also enhances analytical skills, which are invaluable in academic and professional endeavors.
Other exercises in this chapter
Problem 65
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