Simplifying Logarithmic Expressions A Step-by-Step Guide

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Introduction to Logarithmic Simplification

In mathematics, logarithmic expressions often appear complex at first glance, but they can be simplified using various logarithmic properties. These properties allow us to manipulate and combine logarithmic terms, making calculations easier and revealing the underlying structure of the expression. Simplifying logarithmic expressions is a fundamental skill in algebra and calculus, with applications in various fields such as physics, engineering, and computer science. This article delves into simplifying the logarithmic expression 2log 169 - 3log 143 + log 1100 - log 1300 + log 121, providing a step-by-step guide to understanding and applying the necessary logarithmic properties. By mastering these techniques, one can approach more complex problems with confidence and precision.

Before diving into the simplification process, it's crucial to understand the basic logarithmic properties that will be used. The properties include the power rule, product rule, quotient rule, and the change of base formula. The power rule states that log_b(x^p) = p * log_b(x), which allows us to move exponents out of the logarithm. The product rule, log_b(xy) = log_b(x) + log_b(y), helps in combining logarithmic terms with multiplication inside the logarithm. The quotient rule, log_b(x/y) = log_b(x) - log_b(y), deals with division inside the logarithm. Additionally, understanding the change of base formula can be useful in some cases, though it's not strictly needed for this specific problem. Applying these rules systematically is key to simplifying any logarithmic expression effectively. The simplification process not only involves mathematical manipulations but also requires a conceptual understanding of logarithms and their behavior. Therefore, each step in the process will be explained in detail to ensure a clear understanding of the underlying principles.

Breaking Down the Expression: 2log 169 - 3log 143 + log 1100 - log 1300 + log 121

To effectively simplify the given expression, 2log 169 - 3log 143 + log 1100 - log 1300 + log 121, it's crucial to break it down into manageable parts. The expression consists of several logarithmic terms, each with its own coefficient and argument. The initial step involves identifying the prime factors of the numbers inside the logarithms. This helps in applying logarithmic properties more effectively. For example, 169 can be expressed as 13^2, 143 as 11 * 13, 1100 as 11 * 100 or 11 * 10^2, 1300 as 13 * 100 or 13 * 10^2, and 121 as 11^2. Recognizing these prime factorizations is the foundation for using the power, product, and quotient rules of logarithms.

Next, we'll rewrite the expression using these prime factorizations. This will make it easier to apply logarithmic properties. For instance, 2log 169 becomes 2log(13^2), -3log 143 becomes -3log(11 * 13), log 1100 becomes log(11 * 10^2), log 1300 becomes log(13 * 10^2), and log 121 becomes log(11^2). By expressing each number as a product of its prime factors, we prepare the expression for the next phase of simplification. Each term now contains logarithms of products or powers, which can be further simplified using the appropriate logarithmic rules. This process of breaking down and rewriting the expression is essential for simplifying complex logarithmic expressions and revealing their underlying structure.

Applying Logarithmic Properties: Power, Product, and Quotient Rules

Applying logarithmic properties is the key step in simplifying the expression. The power rule, log_b(x^p) = p * log_b(x), is particularly useful for terms like 2log(13^2) and log(11^2). Applying this rule, 2log(13^2) becomes 2 * 2log(13) or 4log(13), and log(11^2) becomes 2log(11). These transformations simplify the terms and make them easier to combine with others.

The next step involves using the product rule, log_b(xy) = log_b(x) + log_b(y), to expand terms like -3log(11 * 13) and log(11 * 10^2) and log(13 * 10^2). Applying this rule, -3log(11 * 13) becomes -3(log(11) + log(13)) or -3log(11) - 3log(13). Similarly, log(11 * 10^2) becomes log(11) + log(10^2), and log(13 * 10^2) becomes log(13) + log(10^2). These expansions break down complex logarithms into simpler terms that can be more easily manipulated.

Finally, we can simplify log(10^2) using the power rule again, resulting in 2log(10). Since log base 10 of 10 is 1, log(10^2) simplifies to 2. Now, combining all the simplified terms, the expression becomes 4log(13) - 3log(11) - 3log(13) + log(11) + 2 - (log(13) + 2) + 2log(11). By carefully applying these logarithmic properties, we've transformed the original expression into a sum of simpler logarithmic terms and constants. The next step is to collect and combine like terms to arrive at the final simplified form.

Combining Like Terms and Final Simplification

After applying logarithmic properties, the expression is now in the form: 4log(13) - 3log(11) - 3log(13) + log(11) + 2 - log(13) - 2 + 2log(11). The next crucial step is to combine the like terms. This involves grouping terms with the same logarithmic component and constant terms together. By doing so, we can further simplify the expression and reveal its final, concise form.

First, let's group the logarithmic terms involving log(13): 4log(13) - 3log(13) - log(13). Combining these, we get (4 - 3 - 1)log(13), which simplifies to 0log(13) or simply 0. Next, we group the logarithmic terms involving log(11): -3log(11) + log(11) + 2log(11). Combining these, we get (-3 + 1 + 2)log(11), which simplifies to 0log(11) or 0 as well. Finally, we combine the constant terms: 2 - 2, which equals 0.

With all like terms combined, the simplified expression is 0log(13) + 0log(11) + 0. Since any term multiplied by 0 is 0, the entire expression simplifies to 0. Thus, the final simplified form of the original expression 2log 169 - 3log 143 + log 1100 - log 1300 + log 121 is 0. This result highlights the power of logarithmic properties in simplifying complex expressions to their most basic form.

Conclusion: Simplified Expression and Key Takeaways

In conclusion, the original expression 2log 169 - 3log 143 + log 1100 - log 1300 + log 121 simplifies to 0. This simplification was achieved through a methodical application of logarithmic properties, including the power rule, product rule, and quotient rule. The process involved breaking down the expression, identifying prime factors, applying relevant logarithmic properties, combining like terms, and arriving at the final simplified result. This exercise underscores the importance of understanding and effectively using logarithmic properties to solve mathematical problems.

The key takeaways from this simplification process are several. First, recognizing the prime factorization of numbers within logarithms is essential for applying logarithmic properties effectively. Second, understanding and correctly applying the power, product, and quotient rules is crucial for manipulating and simplifying logarithmic expressions. Third, the ability to combine like terms is vital for arriving at the final simplified form. Lastly, this exercise demonstrates how complex expressions can be simplified through a systematic and step-by-step approach. By mastering these skills, one can tackle a wide range of logarithmic problems with confidence and precision. Logarithmic simplification is not just a mathematical technique but a powerful tool for solving real-world problems in various scientific and engineering disciplines. The result highlights the elegance and efficiency of logarithmic operations in reducing complex calculations to simpler forms, further emphasizing their importance in mathematical analysis and practical applications.