Problem 65 - Algebra for College Students

Question

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8} 88\)

Step-by-Step Solution

Answer

\( \log_8 88 = 2.1531 \).

Step 1: Apply the Product Rule for Logarithms

Recognize that 88 can be broken down into a product of 8 and 11. Therefore, you can use the product rule for logarithms: \( \log_b (mn) = \log_b m + \log_b n \). Here, set \( m = 8 \) and \( n = 11 \).

Step 2: Apply the Product Rule

Using the product rule, the expression for \( \log_8 88 \) becomes \( \log_8 8 + \log_8 11 \).

Step 3: Evaluate \( \log_8 8 \)

Recognize that the base raised to what power equals itself results in \( 1 \). So, \( \log_8 8 = 1 \).

Step 4: Evaluate \( \log_8 11 \) using Given Value

Use the given value \( \log_8 11 = 1.1531 \), from the problem statement.

Step 5: Add the Logarithms

Combine the values to solve for \( \log_8 88 \): \( 1 + 1.1531 = 2.1531 \). Thus, \( \log_8 88 = 2.1531 \).

Key Concepts

Product Rule

The product rule for logarithms is a handy tool when dealing with the logarithm of a product of two numbers. It states that for any base \( b \), the logarithm of a product \( mn \) can be expressed as the sum of the logarithms of the two numbers individually:
\[ \log_b (mn) = \log_b m + \log_b n \]This property simplifies complex logarithm expressions by breaking them down into simpler parts. In the original exercise, this rule was used by identifying that the number 88 can be expressed as a product of 8 and 11.

• First, it allows one to calculate the logarithm of larger numbers by splitting them into multiplicands with known logarithmic values.
• This method reduces complex calculations.
• The product rule helps in various fields such as calculus and algebra to simplify equations.

By using the product rule, we broke down the problem of finding \( \log_8 88 \) into the simpler problem of finding \( \log_8 8 + \log_8 11 \). This approach simplifies calculations and reveals the strengths of logarithmic properties.

Logarithm Properties

Logarithm properties are fascinating mathematical rules that help simplify and manipulate logarithmic expressions. Apart from the product rule, there are other key properties too, such as the quotient rule and power rule. Understanding these can help solve logarithmic equations efficiently.
- **Power Rule:** This states that \( \log_b (m^n) = n \cdot \log_b m \). It allows you to take a power out in front of the logarithm, simplifying expressions involving exponents.- **Quotient Rule:** It indicates that \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \). This helps in breaking down division inside a logarithm into a subtraction problem.
Returning to our exercise, we predominantly used the product rule for an easier breakdown of the expression \( \log_8 88 = \log_8 8 + \log_8 11 \). By leaning on these rules, logarithmic calculations become far less tedious, allowing for the straightforward solving of problems encountered in calculus, physics, and engineering.
Understanding these properties provides a strong foundation for effectively handling real-world problems where logarithms naturally occur.

Base 8 Logarithm

Logarithms can have different bases, indicating the number we repeatedly multiply. The original exercise utilized a base 8 logarithm, denoted as \( \log_8 \). This means you are considering how many times you need to multiply 8 to reach a given number.
For instance, \( \log_8 8 = 1 \) because 8 raised to the power of 1 equals 8 itself.

• Understanding base-specific logarithms is essential as they appear in different contexts like digital technology and information theory.
• Base 10 (or common logarithm) and base \( e \) (or natural logarithm) are the most frequently used bases in calculations.
• Base 2, known as binary logarithms, is important in computer science for data flow and storage processes.

In our specific solution, knowing the base 8 logarithm helped us efficiently calculate \( \log_8 88 \). This information was combined with provided values like \( \log_8 11 = 1.1531 \) to directly solve the problem using other logarithm principles.