Problem 61 - Algebra for College Students

Question

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8}\left(\frac{5}{11}\right)\)

Step-by-Step Solution

Answer

\( \log_8\left( \frac{5}{11} \right) = -0.3791 \)

Step 1: Understand Properties 10.5-10.7

Properties 10.5-10.7 refer to logarithmic rules such as the quotient rule. The quotient rule states that \( \log_b\left( \frac{M}{N} \right) = \log_b M - \log_b N \). This allows us to convert a logarithm of a division into the difference of two logarithms.

Step 2: Apply the Quotient Rule

Using the quotient rule of logarithms, we can rewrite the given logarithmic expression as \( \log_8\left( \frac{5}{11} \right) = \log_8 5 - \log_8 11 \).

Step 3: Substitute Known Values

We are given \( \log_8 5 = 0.7740 \) and \( \log_8 11 = 1.1531 \). Substitute these values into the expression: \( \log_8 5 - \log_8 11 = 0.7740 - 1.1531 \).

Step 4: Calculate the Result

Perform the subtraction: \( 0.7740 - 1.1531 = -0.3791 \). This is the value of \( \log_8\left( \frac{5}{11} \right) \).

Key Concepts

Quotient Rule in Logarithms

One of the most useful tools in evaluating logarithmic expressions is the quotient rule. The quotient rule essentially allows us to simplify logarithms into more manageable parts. This is particularly helpful when working with divisions inside logarithms.
The quotient rule for logarithms states:
\[\log_b\left( \frac{M}{N} \right) = \log_b M - \log_b N\]This rule helps us express the logarithm of a fraction as the difference between two separate logarithms, which is considerably easier to work with.

• It's particularly effective for substituting and calculating when the values of logs are known separately.
• Translating complicated logarithmic expressions through the quotient rule can greatly simplify problem-solving.

By applying the quotient rule here, we can better understand how to break down and solve logarithmic equations with division.

Understanding Logarithmic Properties

Logarithms have several key properties that make them incredibly useful in mathematics. Three of these important properties (often referenced as Properties 10.5-10.7 in textbooks) include the product rule, the quotient rule, and the power rule. Each of these simplifies complex calculations and elucidates the relationships within logarithms.
For example, the quotient rule we've just discussed allows us to transform the division of numbers inside a logarithm into a much easier subtraction operation. Other properties, like the product rule, express the logarithm of a product as a sum, and the power rule simplifies exponential terms inside logarithms.

• The **product rule:** \( \log_b (MN) = \log_b M + \log_b N \)
• The **quotient rule:** \( \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \)
• The **power rule:** \( \log_b (M^n) = n \cdot \log_b M \)

These properties are essential understandings for anyone diving into logarithmic operations, making expressions more straightforward and solving problems more manageable.

Simplifying Algebraic Expressions with Logarithms

Logarithms become even more powerful when applied to algebraic expressions. They transform often complicated expressions into simpler forms, making analysis and solution-finding more accessible. For instance, using logarithmic properties like the quotient rule can substitute challenging expressions with easier subproblems.
When dealing with algebraic expressions, particularly those involving large numbers or complex fractions, applying logarithmic properties efficiently makes processes like simplification and evaluation more systematic.

• Complex fractions within logarithms become simple differences thanks to the quotient rule.
• The product rule allows combinations of terms to transform into straightforward additions.
• By applying the power rule, exponential components reduce into multipliers.

For students and practitioners, mastering the use of logarithmic properties to tackle algebraic expressions is invaluable, not only for solving textbook problems but also for real-world applications.