Simplifying Logarithmic Expressions A Detailed Guide To 2 Log 2 + Log 5 - 1/2 Log 36 - Log 1/30

Introduction to Logarithmic Expressions

In the realm of mathematics, logarithmic expressions play a pivotal role in simplifying complex calculations and revealing hidden relationships between numbers. This comprehensive guide delves into the intricacies of logarithmic expressions, focusing on the specific example of 2 log 2 + log 5 - 1/2 log 36 - log 1/30. We will dissect this expression step by step, unraveling the underlying principles and techniques involved in its simplification. Our journey will encompass a thorough exploration of the fundamental laws of logarithms, their applications, and the nuances of manipulating logarithmic terms. Whether you're a student grappling with logarithmic equations or a seasoned mathematician seeking a refresher, this exploration promises to enhance your understanding and proficiency in handling logarithmic expressions. By the end of this guide, you will not only be able to simplify the given expression with ease but also possess a robust foundation for tackling a wide array of logarithmic challenges. So, let's embark on this mathematical adventure, demystifying the world of logarithms and unlocking their potential to simplify and illuminate the intricate patterns of numbers.

Understanding the Fundamentals of Logarithms

Before we dive into the intricacies of the expression 2 log 2 + log 5 - 1/2 log 36 - log 1/30, it is crucial to lay a solid foundation by understanding the fundamental principles of logarithms. A logarithm is essentially the inverse operation of exponentiation. To put it simply, if we have an equation like b^x = y, the logarithm of y to the base b is x. Mathematically, this is expressed as log_b(y) = x. Here, 'b' is the base of the logarithm, 'y' is the argument, and 'x' is the exponent to which 'b' must be raised to obtain 'y'. The most commonly used bases are 10 (common logarithm) and e (natural logarithm), denoted as log_10 and ln, respectively. When the base is not explicitly written, it is generally assumed to be 10. Logarithms possess several key properties that are instrumental in simplifying complex expressions. These properties, often referred to as the laws of logarithms, include the product rule, the quotient rule, and the power rule. The product rule states that the logarithm of a product is the sum of the logarithms of the individual factors: log_b(mn) = log_b(m) + log_b(n). The quotient rule states that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator: log_b(m/n) = log_b(m) - log_b(n). The power rule states that the logarithm of a number raised to a power is the product of the power and the logarithm of the number: log_b(m^p) = p log_b(m). These rules are the building blocks for manipulating and simplifying logarithmic expressions, and a thorough understanding of them is essential for mastering logarithmic calculations. As we proceed to tackle the given expression, we will see how these rules come into play, allowing us to break down complex terms into simpler, more manageable components. By grasping the essence of these fundamental principles, we pave the way for a deeper appreciation of the power and elegance of logarithms in mathematical problem-solving.

Breaking Down the Expression: 2 log 2 + log 5 - 1/2 log 36 - log 1/30

Now that we have a firm grasp of the fundamental principles of logarithms, let's embark on the journey of dissecting and simplifying the expression 2 log 2 + log 5 - 1/2 log 36 - log 1/30. This expression presents a mix of logarithmic terms, each with its own coefficient and argument, requiring a systematic approach to unravel its complexities. Our strategy will involve applying the laws of logarithms, which we discussed earlier, to manipulate and combine these terms. The first term, 2 log 2, can be simplified using the power rule of logarithms, which states that log_b(m^p) = p log_b(m). Applying this rule in reverse, we can rewrite 2 log 2 as log 2^2, which simplifies to log 4. This transformation is a crucial step in consolidating the expression. Next, we examine the term - 1/2 log 36. Again, employing the power rule, we can express this as log 36^(-1/2). Recognizing that a negative exponent indicates a reciprocal and a fractional exponent represents a root, we can further simplify 36^(-1/2) as 1/√36, which equals 1/6. Thus, - 1/2 log 36 becomes log (1/6). This step demonstrates the power of logarithmic rules in converting coefficients into exponents, making the expression more amenable to further simplification. The final term, - log 1/30, can be handled similarly. We can rewrite this as log (1/30)^(-1). A negative exponent implies a reciprocal, so (1/30)^(-1) is simply 30. Therefore, - log 1/30 transforms into log 30. By systematically applying the power rule, we have successfully reshaped the expression, paving the way for further simplification using the product and quotient rules. This meticulous breakdown highlights the importance of mastering logarithmic laws in effectively manipulating complex logarithmic expressions.

Applying Logarithmic Laws for Simplification

With the expression now transformed into log 4 + log 5 - 1/2 log 36 - log 1/30, our next step involves strategically applying the laws of logarithms to further simplify and consolidate the terms. We've already used the power rule to transform the coefficients into exponents, and now we'll leverage the product and quotient rules to combine the logarithmic terms. Recall that the product rule states that log_b(mn) = log_b(m) + log_b(n), and the quotient rule states that log_b(m/n) = log_b(m) - log_b(n). These rules provide the framework for combining logarithmic terms with addition and subtraction operations. Let's begin by focusing on the first two terms: log 4 + log 5. According to the product rule, the sum of two logarithms is equal to the logarithm of the product of their arguments. Therefore, log 4 + log 5 can be rewritten as log (4 * 5), which simplifies to log 20. This transformation effectively reduces two terms into a single term, moving us closer to the final simplified form. Next, we need to incorporate the term log (1/6), which we derived from - 1/2 log 36, and the term log 30, which we obtained from - log 1/30. The expression now looks like log 20 + log (1/6) + log 30. Applying the product rule again, we can combine log 20 and log (1/6) as log (20 * 1/6), which simplifies to log (10/3). The expression now becomes log (10/3) + log 30. One final application of the product rule allows us to combine these two terms: log ((10/3) * 30). This simplifies to log 100. This strategic application of the product rule has effectively merged the individual logarithmic terms into a single, more manageable term. The next step will involve evaluating this final logarithmic expression, bringing us to the culmination of our simplification process. This systematic approach, guided by the laws of logarithms, demonstrates the power of these rules in transforming complex expressions into simpler, more understandable forms.

Evaluating the Final Logarithmic Expression

Having successfully simplified the original expression 2 log 2 + log 5 - 1/2 log 36 - log 1/30 to log 100, the final step in our journey is to evaluate this logarithmic expression. Evaluating a logarithm essentially means finding the exponent to which the base must be raised to obtain the argument. In this case, we have log 100. As we discussed earlier, when the base of the logarithm is not explicitly written, it is generally assumed to be 10. Therefore, log 100 is equivalent to log_10(100). The question we need to answer is: to what power must we raise 10 to obtain 100? We know that 10 squared (10^2) is equal to 100. Thus, log_10(100) = 2. This evaluation provides the final, simplified value of the original expression. By systematically applying the laws of logarithms and breaking down the expression into manageable steps, we have successfully navigated the complexities of logarithmic manipulation and arrived at a concise numerical answer. This process underscores the importance of understanding logarithmic properties and their strategic application in simplifying mathematical expressions. The journey from the initial expression to the final value showcases the elegance and power of logarithms in revealing the underlying relationships between numbers. This comprehensive evaluation not only provides the solution but also reinforces the fundamental principles of logarithmic calculations, equipping us with the skills to tackle a wide range of logarithmic problems with confidence and precision. The final answer, 2, represents the culmination of our efforts and a testament to the transformative power of logarithmic simplification.

Conclusion: Mastering Logarithmic Simplification

In conclusion, the process of simplifying the logarithmic expression 2 log 2 + log 5 - 1/2 log 36 - log 1/30 has been a comprehensive exploration of the fundamental principles and techniques involved in logarithmic manipulation. We began by establishing a solid understanding of the definition of logarithms and the crucial laws that govern their behavior, including the power rule, product rule, and quotient rule. These laws served as our guiding tools throughout the simplification process, allowing us to break down the complex expression into manageable components. We meticulously applied the power rule to transform coefficients into exponents, effectively reshaping the terms and paving the way for further simplification. Subsequently, we strategically employed the product and quotient rules to combine logarithmic terms, reducing the expression to a single logarithmic term: log 100. Finally, we evaluated this expression, determining the exponent to which the base (10, in this case) must be raised to obtain the argument (100), arriving at the final answer of 2. This journey underscores the importance of a systematic approach when tackling logarithmic expressions. Each step, from understanding the foundational principles to applying the specific laws, plays a critical role in arriving at the simplified solution. Mastering these techniques not only enables us to solve specific problems but also cultivates a deeper understanding of the nature of logarithms and their applications in various mathematical contexts. The ability to simplify logarithmic expressions is a valuable skill, not just in mathematics but also in fields such as physics, engineering, and computer science, where logarithms are used to model and analyze a wide range of phenomena. By diligently practicing and applying these principles, one can develop proficiency in logarithmic simplification and unlock the power of logarithms in solving complex problems.