Evaluating Log₇ 98 A Step-by-Step Guide

In this comprehensive guide, we will delve into the step-by-step process of evaluating the logarithmic expression log₇ 98, given that log₇ 2 ≈ 0.356. Logarithms, often perceived as complex mathematical entities, play a pivotal role in various scientific and engineering disciplines. Understanding how to manipulate and evaluate logarithmic expressions is crucial for anyone seeking a solid foundation in mathematics. We will embark on a journey to dissect the intricacies of this problem, unraveling the underlying principles and techniques that govern logarithmic operations. By the end of this exploration, you will not only be able to evaluate log₇ 98 but also gain a deeper appreciation for the elegance and power of logarithms. This article aims to provide a clear, concise, and accessible explanation, ensuring that readers of all backgrounds can grasp the concepts and apply them effectively. So, let's dive in and conquer the world of logarithms together!

Rewriting log₇ 98 Using the Product Property

The product property of logarithms is a fundamental rule that allows us to simplify logarithmic expressions involving the product of two or more numbers. This property states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, it can be expressed as: logₐ (xy) = logₐ x + logₐ y, where 'a' is the base of the logarithm, and x and y are positive numbers. This property is an invaluable tool in simplifying complex logarithmic expressions and making them easier to evaluate. In the context of our problem, we need to rewrite log₇ 98 using the product property. To achieve this, we must first identify the factors of 98. The prime factorization of 98 is 2 × 7 × 7, which can be written as 2 × 49. Therefore, we can rewrite log₇ 98 as log₇ (2 × 49). Now, applying the product property, we get log₇ (2 × 49) = log₇ 2 + log₇ 49. This transformation is the key to solving the problem, as it breaks down the original expression into simpler terms that we can evaluate using the given information and basic logarithmic properties. Understanding and applying the product property is crucial for manipulating and simplifying logarithmic expressions, paving the way for solving more complex problems in mathematics and other fields.

Which of the following correctly rewrites log₇ 98 using the product property?

A. 7 log 2 + 7 log 49 B. log 7 + log 2 + log 49 C. log₇ 2 + log₇ 49

The correct answer is C. log₇ 2 + log₇ 49. This option directly applies the product property of logarithms, accurately expressing log₇ 98 as the sum of the logarithms of its factors, 2 and 49, with the same base 7. Options A and B are incorrect because they either introduce coefficients incorrectly or change the base of the logarithms, which violates the fundamental principles of logarithmic operations. Understanding the product property and its correct application is essential for simplifying and evaluating logarithmic expressions effectively.

Evaluating log₇ 49

Having successfully applied the product property to rewrite log₇ 98 as log₇ 2 + log₇ 49, our next crucial step is to evaluate log₇ 49. This evaluation is pivotal in determining the final value of the original expression. To evaluate log₇ 49, we need to recall the fundamental definition of a logarithm. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. In other words, logₐ b = x if and only if aˣ = b. Applying this definition to our problem, we need to find the exponent to which 7 must be raised to obtain 49. We know that 7² = 49. Therefore, log₇ 49 = 2. This simple yet powerful evaluation is a cornerstone of logarithmic operations. By recognizing the relationship between the base and the argument of the logarithm, we can directly determine its value. This step not only provides a numerical value for log₇ 49 but also reinforces our understanding of the fundamental principles of logarithms. Mastering this skill is crucial for tackling more complex logarithmic problems and applications in various fields.

log₇ 49 = 2

Putting It All Together: Evaluating log₇ 98

With the individual components of our expression evaluated, we now have all the necessary pieces to determine the final value of log₇ 98. We have established that log₇ 98 can be rewritten as log₇ 2 + log₇ 49 using the product property. We also know that log₇ 2 ≈ 0.356 (given) and log₇ 49 = 2 (calculated). To find the value of log₇ 98, we simply need to add these two values together: log₇ 98 = log₇ 2 + log₇ 49 ≈ 0.356 + 2 = 2.356. This final step demonstrates the power of breaking down complex problems into smaller, manageable parts. By applying the product property and evaluating individual logarithmic terms, we have successfully calculated the value of log₇ 98. This process highlights the importance of understanding and applying logarithmic properties and definitions. The ability to manipulate and evaluate logarithmic expressions is a valuable skill in various mathematical and scientific contexts. Through this step-by-step approach, we have not only solved the problem but also reinforced our understanding of the underlying principles of logarithms.

Therefore, log₇ 98 ≈ 2.356.

Conclusion

In this comprehensive guide, we have meticulously dissected the process of evaluating log₇ 98, given that log₇ 2 ≈ 0.356. We began by exploring the fundamental product property of logarithms, which allowed us to rewrite log₇ 98 as log₇ 2 + log₇ 49. This crucial step transformed the original expression into a more manageable form. We then delved into the evaluation of log₇ 49, leveraging the definition of a logarithm to determine that log₇ 49 = 2. Finally, we combined these individual components, adding the values of log₇ 2 and log₇ 49 to arrive at the final answer: log₇ 98 ≈ 2.356. This journey through logarithmic manipulation and evaluation has underscored the importance of understanding and applying logarithmic properties and definitions. The ability to break down complex problems into simpler steps is a hallmark of effective problem-solving in mathematics and beyond. By mastering these techniques, you can confidently tackle a wide range of logarithmic challenges and appreciate the elegance and power of these mathematical tools. We encourage you to continue exploring the world of logarithms, delving deeper into their applications and nuances. The knowledge and skills you have gained here will serve as a solid foundation for further mathematical endeavors.