Problem 60 - Algebra for College Students

Question

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8} 55\)

Step-by-Step Solution

Answer

\( \log_8 55 = 1.9271 \)

Step 1: Identify the Property

We need to evaluate \ \( \log_8 55 \). We can break down \( 55 \) into its prime factors, \( 55 = 5 \times 11 \). We will use the property \( \log_b (MN) = \log_b M + \log_b N \) to separate the log of the product.

Step 2: Apply the Logarithm Property

Using the property from Step 1, \[ \log_8 55 = \log_8 (5 \times 11) = \log_8 5 + \log_8 11. \]

Step 3: Substitute Provided Values

We are given \( \log_8 5 = 0.7740 \) and \( \log_8 11 = 1.1531 \). Substitute these values into the expression from Step 2:\[ \log_8 55 = 0.7740 + 1.1531. \]

Step 4: Perform the Addition

Add the two logarithmic values:\[ 0.7740 + 1.1531 = 1.9271. \]

Key Concepts

Logarithmic Properties

Logarithmic properties are incredibly useful tools that simplify the process of working with logarithms. One fundamental property is the Product Property:

• When you have a logarithm of a product, such as \( \log_b (MN) \), you can separate it into a sum of two logarithms: \( \log_b M + \log_b N \).
• This property is derived from the concept that logarithms convert multiplication into addition, which is a defining characteristic of logarithms.

To see this property in action, consider the expression \( \log_8 55 \). Since 55 can be represented as a product of its prime factors (5 and 11), you can use the product property to break it down: \[ \log_8 55 = \log_8 (5 \times 11) = \log_8 5 + \log_8 11. \]By breaking down the logarithm into smaller parts, calculations become more manageable.

Evaluating Logarithms

Evaluating logarithms might seem complex at first, but it becomes straightforward with practice and familiarity with logarithmic properties.

• The goal of evaluating a logarithm is to find its value, which often represents an exponent in exponential form.
• When given partial logarithmic values as in the exercise, you substitute these known values into the expression.

For example, in the exercise, you are given \( \log_8 5 = 0.7740 \) and \( \log_8 11 = 1.1531 \). These values are used directly in the simplified expression from the property:Replace \( \log_8 5 \) and \( \log_8 11 \) with their given numerical values:\[ \log_8 55 = \log_8 5 + \log_8 11 = 0.7740 + 1.1531. \]By performing the addition, you determine that \( \log_8 55 = 1.9271 \), completing the evaluation.

Prime Factorization

Prime factorization is the process of expressing a whole number as a product of its prime numbers. Understanding the prime factorization of a number is crucial when using logarithmic properties, especially when dealing with products inside a logarithm.

• Every number greater than 1 can be uniquely factored into primes, which are numbers divisible only by 1 and themselves.
• With prime factorization, you simplify complex logarithmic expressions into manageable parts.

Take 55 for instance. It can be decomposed into its prime components as follows:\[ 55 = 5 \times 11 \]Both 5 and 11 are prime numbers. Knowing this factorization is essential when applying the logarithmic product property to evaluate the logarithm of 55. As you've seen, using these distinctive factors, the complex logarithm is broken into simpler parts: \( \log_8 5 \) and \( \log_8 11 \). By using these individual logarithms, the calculation becomes straightforward, and prime factorization thus plays a key supportive role in evaluating logarithms.