Evaluate Log10 108 Given Log10 2 = 0.3010, Log10 3 = 0.4771, And Log10 5 = 0.6990

Introduction to Logarithms and Their Properties

In the realm of mathematics, logarithms serve as a powerful tool for simplifying complex calculations and solving exponential equations. Understanding the fundamental properties of logarithms is crucial for effectively manipulating and evaluating logarithmic expressions. This article delves into the evaluation of $log_{10} 108$ given the values of $log_{10} 2$, $log_{10} 3$, and $log_{10} 5$. We will leverage key logarithmic properties such as the product rule, power rule, and the relationship between different bases to arrive at the solution. The principles discussed here not only aid in solving this specific problem but also provide a solid foundation for tackling a wide array of logarithmic challenges. Before diving into the solution, let's recap the essential properties of logarithms that will be instrumental in our calculations. These include the product rule, which states that the logarithm of a product is the sum of the logarithms of the individual factors; the power rule, which allows us to simplify logarithms of powers; and the change of base formula, which enables us to convert logarithms from one base to another. These properties are the building blocks for manipulating logarithmic expressions and will be extensively used throughout this article.

Problem Statement: Evaluating $log_{10} 108$

Our objective is to determine the value of $log_{10} 108$, given that $log_{10} 2 = 0.3010$, $log_{10} 3 = 0.4771$, and $log_{10} 5 = 0.6990$. This problem exemplifies how logarithmic properties can be applied to simplify complex expressions into manageable components. The number 108, while seemingly straightforward, can be broken down into its prime factors, allowing us to express $log_{10} 108$ in terms of the known logarithms of 2 and 3. This process highlights the practical application of prime factorization in simplifying logarithmic calculations. We will utilize the prime factorization of 108 to rewrite the expression and then apply logarithmic identities to find the solution. This method demonstrates the elegance and efficiency of logarithms in simplifying calculations that would otherwise be cumbersome. Understanding how to decompose numbers into their prime factors and apply logarithmic properties is a crucial skill in mathematics, particularly in areas such as calculus, physics, and engineering. In the following sections, we will systematically break down 108 and apply the necessary logarithmic properties to arrive at the final answer.

Prime Factorization of 108

The initial step in evaluating $log_{10} 108$ involves finding the prime factorization of 108. This process breaks down 108 into a product of its prime factors, which are the smallest integers that can divide 108 without leaving a remainder. Prime factorization is a fundamental concept in number theory and plays a crucial role in simplifying many mathematical expressions. To find the prime factors of 108, we can start by dividing it by the smallest prime number, 2. We find that 108 divided by 2 is 54. We then divide 54 by 2 again, resulting in 27. Since 27 is not divisible by 2, we move to the next prime number, 3. Dividing 27 by 3 gives us 9, and dividing 9 by 3 gives us 3. Finally, dividing 3 by 3 gives us 1, indicating that we have reached the end of the factorization process. Therefore, the prime factorization of 108 is $2 imes 2 imes 3 imes 3 imes 3$, which can be written as $2^2 imes 3^3$. This prime factorization is the key to expressing $log_{10} 108$ in terms of $log_{10} 2$ and $log_{10} 3$, which are given in the problem statement. The ability to perform prime factorization quickly and accurately is a valuable skill in mathematics, and it is essential for simplifying logarithmic expressions and solving various mathematical problems. In the next section, we will use this prime factorization to rewrite $log_{10} 108$ using logarithmic properties.

Applying Logarithmic Properties

Having determined the prime factorization of 108 as $2^2 imes 3^3$, we can now express $log_{10} 108$ in terms of $log_{10} 2$ and $log_{10} 3$. This step involves applying the fundamental properties of logarithms, specifically the product and power rules. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In mathematical terms, $log_b (xy) = log_b x + log_b y$. Applying this rule to $log_{10} (2^2 imes 3^3)$, we get: $log_{10} 108 = log_{10} (2^2 imes 3^3) = log_{10} (2^2) + log_{10} (3^3)$. Next, we apply the power rule of logarithms, which states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. In mathematical terms, $log_b (x^p) = p imes log_b x$. Applying this rule to our expression, we get: $log_{10} (2^2) + log_{10} (3^3) = 2 imes log_{10} 2 + 3 imes log_{10} 3$. Now, we have expressed $log_{10} 108$ in terms of $log_{10} 2$ and $log_{10} 3$, which are given values. This transformation is a critical step in solving the problem, as it allows us to substitute the given values and calculate the final answer. The ability to apply these logarithmic properties effectively is essential for simplifying complex logarithmic expressions and solving various mathematical problems. In the next section, we will substitute the given values and compute the final result.

Substituting Values and Computing the Result

With the expression $log_{10} 108$ rewritten as $2 imes log_{10} 2 + 3 imes log_{10} 3$, we can now substitute the given values of $log_{10} 2$ and $log_{10} 3$ to compute the final result. We are given that $log_{10} 2 = 0.3010$ and $log_{10} 3 = 0.4771$. Substituting these values into our expression, we get: $2 imes log_{10} 2 + 3 imes log_{10} 3 = 2 imes 0.3010 + 3 imes 0.4771$. Now, we perform the multiplication: $2 imes 0.3010 = 0.6020$ and $3 imes 0.4771 = 1.4313$. Adding these results together, we get: $0.6020 + 1.4313 = 2.0333$. Therefore, $log_{10} 108 = 2.0333$. This calculation demonstrates the power of logarithms in simplifying complex calculations. By breaking down 108 into its prime factors and applying logarithmic properties, we were able to express $log_{10} 108$ in terms of known values and easily compute the result. This method is widely used in various fields, including science, engineering, and finance, where logarithmic calculations are common. The ability to perform these calculations accurately and efficiently is a valuable skill in many disciplines. In the final section, we will summarize our findings and discuss the importance of understanding logarithmic properties.

Conclusion: Importance of Logarithmic Properties

In this article, we successfully evaluated $log_{10} 108$ given the values of $log_{10} 2$, $log_{10} 3$, and $log_{10} 5$. We began by understanding the fundamental properties of logarithms, including the product rule and the power rule. We then broke down 108 into its prime factors, expressing it as $2^2 imes 3^3$. Applying the product and power rules of logarithms, we rewrote $log_{10} 108$ as $2 imes log_{10} 2 + 3 imes log_{10} 3$. Finally, we substituted the given values and computed the result, finding that $log_{10} 108 = 2.0333$. This exercise highlights the importance of understanding logarithmic properties in simplifying complex mathematical expressions. Logarithms are not just abstract mathematical concepts; they are powerful tools with practical applications in various fields. From calculating the magnitude of earthquakes to modeling population growth, logarithms play a crucial role in many scientific and engineering disciplines. The ability to manipulate logarithmic expressions and solve logarithmic equations is a valuable skill for anyone pursuing a career in these fields. Furthermore, logarithms are used extensively in computer science, finance, and statistics, making a solid understanding of their properties essential for success in these areas as well. By mastering logarithmic properties, individuals can enhance their problem-solving skills and tackle a wide range of mathematical challenges with confidence. This article serves as a testament to the power and versatility of logarithms, and we encourage readers to continue exploring their applications in various contexts.