Calculate Log 5: A Step-by-Step Guide With Examples

Hey guys! Today, let's dive into the fascinating world of logarithms and tackle a common question: How do we calculate log 5 given that log 2 = 0.3010 and log 3 = 0.4771? We'll explore the concepts of change of base and logarithmic properties, making sure you understand each step along the way. So, grab your thinking caps, and let's get started!

Understanding the Fundamentals of Logarithms

Before we jump into the calculation, let's quickly recap what logarithms are all about. At its heart, a logarithm answers the question: "To what power must we raise a base number to get a specific value?" For instance, the logarithm of 100 to base 10 (written as log₁₀ 100) is 2 because 10 raised to the power of 2 equals 100 (10² = 100). In our case, we're dealing with common logarithms, which have a base of 10. When the base isn't explicitly written (like in log 2), it's understood to be 10.

Logarithms are a fundamental concept in mathematics with wide-ranging applications, especially in fields like science and engineering. They provide a method to simplify complex calculations, particularly those involving multiplication, division, and exponentiation. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Mathematically, if b^y = x, then log_b(x) = y, where b is the base, x is the number, and y is the logarithm. Understanding logarithms is crucial because they transform multiplication problems into addition, division problems into subtraction, and exponentiation into multiplication, thereby simplifying calculations. The properties of logarithms, including the product rule, quotient rule, and power rule, enable us to manipulate logarithmic expressions effectively. The product rule states that the logarithm of the product of two numbers is the sum of their logarithms: log_b(mn) = log_b(m) + log_b(n). The quotient rule states that the logarithm of the quotient of two numbers is the difference of their logarithms: log_b(m/n) = log_b(m) - log_b(n). The power rule states that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number: log_b(m^k) = k * log_b(m). These properties are essential tools for simplifying logarithmic equations and solving complex problems. Moreover, the change of base formula allows us to convert logarithms from one base to another, which is particularly useful when calculators or logarithmic tables are only available for certain bases. The change of base formula is given by: log_b(a) = log_c(a) / log_c(b). This formula enables us to express any logarithm in terms of common logarithms (base 10) or natural logarithms (base e), making calculations more manageable and consistent. In the context of our problem, we use the properties of logarithms and the change of base formula to express log 5 in terms of log 2 and other known logarithmic values, demonstrating how these foundational concepts come together to solve practical mathematical questions. Logarithms are not just abstract mathematical tools; they are integral to solving real-world problems, making their comprehension a valuable asset. By mastering these concepts and practicing their application, you enhance your analytical skills and gain a deeper appreciation for the power and utility of mathematics.

The Power of Logarithmic Properties

Logarithms come with a set of handy properties that make calculations easier. The key properties we'll use today are:

1. Product Rule: log (a * b) = log a + log b
2. Quotient Rule: log (a / b) = log a - log b

These properties allow us to break down complex logarithms into simpler ones. Remember these, they're super important!

Change of Base: A Crucial Technique

The change of base is a fundamental concept in dealing with logarithms, allowing us to express logarithms in different bases, which is particularly useful when we want to compute logarithms using calculators or tables that operate in a specific base. In the context of our problem, the change of base is essential because we're given the values of log 2 and log 3, which are common logarithms (base 10), and we need to find log 5, also in base 10. The change of base formula is given by: log_b(a) = log_c(a) / log_c(b), where a is the number whose logarithm we want to find, b is the original base, and c is the new base we want to convert to. This formula allows us to convert logarithms from any base b to any other base c, as long as we know the logarithms in the new base. The flexibility provided by the change of base formula is critical because it enables us to leverage the properties of logarithms and known logarithmic values to compute logarithms that might not be directly available. For instance, if we have a calculator that only calculates base 10 logarithms, we can use the change of base formula to calculate logarithms in any base. In our scenario, we are working with common logarithms (base 10), so the change of base formula might seem less directly applicable since we don't need to change the base. However, the underlying principle of expressing a logarithm in terms of known logarithms is effectively what we are doing. We are using the properties of logarithms to rewrite log 5 in terms of logarithms that we can calculate using the given values of log 2 and log 3. The strategy involves recognizing that 5 can be expressed as 10/2, and then using the quotient rule of logarithms to break down log 5 into log 10 - log 2. Since log 10 is 1 (because 10^1 = 10), we can easily compute log 5 using the given value of log 2. This approach demonstrates the power and utility of understanding logarithmic properties and how they can be applied to solve problems. The change of base formula, while not explicitly used here to change the base, exemplifies the broader idea of manipulating logarithms to make calculations feasible. By mastering this technique, you can tackle a wide range of logarithmic problems with greater confidence and efficiency.

Calculating log 5: Step-by-Step

Here's where the magic happens! We need to express 5 in terms of 2 and 3, or numbers related to 10 (since we're working with base-10 logarithms). The key insight is recognizing that 5 = 10 / 2. Now, we can use the quotient rule:

log 5 = log (10 / 2)

Applying the quotient rule, we get:

log 5 = log 10 - log 2

We know that log 10 (base 10) is 1, and we're given that log 2 = 0.3010. So,

log 5 = 1 - 0.3010

log 5 = 0.6990

And there you have it! The value of log 5 is 0.6990.

This method highlights the importance of understanding logarithmic properties and how they can be applied to simplify calculations. By breaking down complex expressions into simpler components, we can solve problems that initially seem daunting. The ability to manipulate logarithmic expressions is a valuable skill in mathematics and its applications. The simplicity of this calculation belies the power of the underlying principles. By recognizing the relationship between 5, 10, and 2, we could transform a seemingly difficult problem into a straightforward subtraction. This approach underscores the essence of mathematical problem-solving: identifying key relationships and leveraging them to find solutions. Moreover, this exercise reinforces the importance of memorizing basic logarithmic values, such as log 10, and understanding the fundamental properties of logarithms. These are the building blocks upon which more complex calculations are based. In the context of exams and practical applications, being able to quickly apply these properties can save time and improve accuracy. This example also illustrates the interconnectedness of mathematical concepts. Logarithms, division, and subtraction all come together in this solution, demonstrating how a solid understanding of foundational topics can lead to mastery of more advanced concepts. By practicing similar problems and internalizing these techniques, you'll develop a robust toolkit for tackling logarithmic challenges. The result, log 5 = 0.6990, is not just a numerical answer; it's a testament to the power of mathematical reasoning and the elegance of logarithmic transformations.

Choosing the Correct Option

Looking at the options provided:

A) 0.6990 B) 0.6991 C) 0.6992 D) 0.6993

The correct answer is A) 0.6990.

Alternatives and Other Methods

While we've calculated log 5 using the quotient rule, there are other ways we could have approached this. For example, if we knew the value of log 100 (which is 2), we could express 5 as the square root of 25 and use the power rule of logarithms, but our method is the most straightforward given the information we had. Remember, the beauty of mathematics often lies in the multiple paths you can take to reach the same destination. Understanding different approaches not only strengthens your problem-solving skills but also deepens your comprehension of the underlying concepts. In this case, while the method we used—expressing 5 as 10/2 and applying the quotient rule—is the most efficient given the available information, exploring alternative methods can provide additional insights. For instance, if we were to consider the relationship between 5 and other numbers whose logarithms we know, such as 100 (since log 100 = 2), we could explore expressing 5 as a factor or a root of 100. However, this approach might involve more complex calculations or the use of additional logarithmic properties. Another alternative method could involve using the fact that 5 is the square root of 25. If we knew log 25, we could apply the power rule of logarithms to find log 5. However, this would require an additional step of finding log 25, which might not be as straightforward as our initial method. The efficiency of the quotient rule method stems from its direct application to the given information: we knew log 2 and log 10, and the relationship 5 = 10/2 allowed us to leverage these values immediately. This highlights the importance of choosing the most appropriate method based on the problem's specific details and the available information. Furthermore, understanding multiple methods for solving the same problem can be beneficial in various ways. It allows you to verify your answer using a different approach, ensuring accuracy. It also enhances your problem-solving versatility, enabling you to adapt to different problem structures and constraints. In the broader context of mathematical education, encouraging the exploration of alternative methods fosters creativity and deeper understanding. It moves beyond rote memorization of formulas and procedures, promoting a more flexible and intuitive approach to problem-solving. By engaging with different solution pathways, students develop a richer appreciation for the interconnectedness of mathematical concepts and the power of mathematical reasoning.

Final Thoughts

So, there you have it! We've successfully calculated log 5 using the properties of logarithms and a little bit of clever thinking. This is just one example of how understanding the fundamentals can unlock a whole world of mathematical possibilities. Keep practicing, and you'll be a logarithm whiz in no time!

Remember, the key takeaway from this exercise is the power of logarithmic properties and the strategic application of these properties to solve problems. By mastering these techniques, you can tackle a wide range of logarithmic challenges with greater confidence and efficiency. The specific method we used, expressing 5 as 10/2 and applying the quotient rule, is a prime example of how a well-chosen strategy can simplify a seemingly complex problem. This approach underscores the importance of careful analysis and planning before diving into calculations. In the world of mathematics, there are often multiple pathways to a solution, but some paths are more direct and efficient than others. Developing the ability to identify the most appropriate method is a crucial skill for success in mathematical problem-solving. Furthermore, this exercise highlights the importance of having a solid foundation in the fundamental concepts of logarithms. Understanding the properties of logarithms, such as the product rule, quotient rule, and power rule, is essential for manipulating logarithmic expressions and solving equations. Without this foundational knowledge, it becomes much more difficult to tackle more advanced problems. The practical implications of this knowledge extend beyond the classroom. Logarithms are used in various fields, including science, engineering, and finance. Understanding logarithms is essential for analyzing data, modeling phenomena, and making informed decisions in these areas. By mastering the concepts and techniques presented in this guide, you are not only improving your mathematical skills but also preparing yourself for success in a wide range of academic and professional pursuits. The joy of problem-solving often comes from the satisfaction of finding a clever solution to a challenging problem. In this case, the elegance and simplicity of the quotient rule method make the solution particularly satisfying. By continuing to practice and explore mathematical concepts, you will develop the skills and intuition needed to discover these elegant solutions on your own.