Problem 60 - Intermediate Algebra

Question

For Problems \(60-68\), you are given \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\). Evaluate each expression using Properties 11.5-11.7. $$ \log _{8} 55 $$

Step-by-Step Solution

Answer

\( \log_{8} 55 = 1.9271 \).

Step 1: Use the Product Property of Logarithms

Recall the product property of logarithms, which states that \( \log_{b}(MN) = \log_{b}M + \log_{b}N \). Here, we can express 55 as a product: 55 = 5 \times 11.

Step 2: Apply the Product Property to the Expression

Using the product property, we rewrite the given expression: \( \log_{8} 55 = \log_{8} (5 \times 11) \). This becomes \( \log_{8} 5 + \log_{8} 11 \).

Step 3: Substitute the Given Values

Substitute the supplied logarithm values: \( \log_{8} 5 = 0.7740 \) and \( \log_{8} 11 = 1.1531 \). Thus, \( \log_{8} 55 = 0.7740 + 1.1531 \).

Step 4: Perform the Addition

Add the numbers obtained from substitution: 0.7740 + 1.1531 = 1.9271. Thus, \( \log_{8} 55 = 1.9271 \).

Key Concepts

Product Property of Logarithms

Logarithms have several useful properties that make complex expressions simpler to evaluate. One of the most handy properties in this toolkit is the product property, which elegantly transforms log expressions involving products. The product property of logarithms states that:

• For any positive numbers M and N, and a base b, the logarithm of a product is the sum of the logarithms of the factors: \[ \log_{b}(MN) = \log_{b}M + \log_{b}N \]

This property is especially helpful when dealing with logarithms where direct calculation isn't straightforward. It cuts down a potentially difficult problem into simpler, smaller parts. In our example, we transformed the product within a logarithm into a sum of two logarithmic evaluations we already know. Breaking things down makes the process easier and avoids unnecessary complications.

Logarithmic Evaluation

Logarithmic evaluation involves finding the numerical value of a logarithm. When given specific values as in our exercise, it’s a straightforward process of substitution.

• In the example provided, we needed to evaluate \( \log_{8} 55 \).
• We did not have this value directly, but we did have \( \log_{8} 5 = 0.7740 \) and \( \log_{8} 11 = 1.1531 \).

Using the product property of logarithms, we expressed 55 as 5 times 11, turning a single logarithmic expression into a combination of known values.
This step transforms the evaluation into a simple arithmetic problem: adding the two known logarithm results. This approach highlights the beauty of logarithms in mathematical problem-solving: leveraging known values to find unknowns.
Thus, the evaluation becomes a matter of basic addition, simplifying the overall task.

Mathematical Problem Solving

Mathematical problem solving is a skill that involves not only understanding formulas and constants but also knowing when and how to apply them. Engaging with problems like this requires strategic thinking.

• Consider the information you have and how it relates to the expression you need to evaluate.
• Employ properties, such as the product property of logarithms, to rewrite expressions in terms of known values.
• Always substitute and simplify in steps to avoid mistakes and keep calculations manageable.

Every problem has its unique angles, and recognizing these can aid in deciding which logarithmic property or mathematical technique will most effectively simplify and solve the problem. Mathematical problem solving is about making the complex straightforward, using logical steps to guide you to the correct answer while ensuring each step is documented clearly.