Expressing Log Base 10 Of 24 In Terms Of M And N

In mathematics, logarithms are a fundamental concept, particularly useful for simplifying complex calculations and understanding exponential relationships. This article delves into the application of logarithmic properties to express the logarithm of a composite number in terms of the logarithms of its prime factors. Specifically, we will explore how to express ${\log_{10} 24}$ in terms of ${m}$ and ${n}$, given that ${\log_{10} 2 = m}$ and ${\log_{10} 3 = n}$. This problem highlights the utility of logarithmic identities in breaking down complex expressions into simpler components. The core idea revolves around leveraging the properties of logarithms, such as the product rule and the power rule, to decompose the number 24 into its prime factors and then express its logarithm in terms of ${m}$ and ${n}$. Understanding these logarithmic manipulations is crucial for various applications in mathematics, engineering, and computer science, where logarithmic scales and transformations are frequently employed to simplify complex problems and reveal underlying patterns. Throughout this discussion, we will emphasize clarity and precision to ensure that readers of all backgrounds can follow the logical progression and grasp the underlying concepts. By the end of this article, you will have a solid understanding of how to manipulate logarithmic expressions and apply them to solve similar problems, enhancing your problem-solving skills in mathematics.

Given that ${\log_{10} 2 = m}$ and ${\log_{10} 3 = n}$, our objective is to find the value of ${\log_{10} 24}$ in terms of ${m}$ and ${n}$. This problem is a classic example of how logarithmic properties can be used to simplify expressions and relate the logarithms of different numbers. The number 24 is not a prime number, and thus, we can decompose it into its prime factors. This decomposition is the key to expressing ${\log_{10} 24}$ in terms of ${\log_{10} 2}$ and ${\log_{10} 3}$, which are given as ${m}$ and ${n}$ respectively. The approach involves breaking down 24 into its prime factors, applying the properties of logarithms to rewrite the expression, and then substituting the given values of ${m}$ and ${n}$. This process not only demonstrates the practical application of logarithmic identities but also reinforces the importance of prime factorization in simplifying mathematical expressions. Understanding how to manipulate logarithms in this way is essential for more advanced topics in mathematics and other quantitative fields. Furthermore, this problem serves as a valuable exercise in algebraic manipulation and logical reasoning, skills that are crucial for success in mathematical problem-solving. By carefully applying the rules of logarithms and performing the necessary substitutions, we can arrive at a solution that expresses ${\log_{10} 24}$ concisely in terms of ${m}$ and ${n}$.

The first step in solving this problem is to determine the prime factorization of 24. Prime factorization is the process of expressing a number as the product of its prime factors. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11). To find the prime factorization of 24, we can use a factor tree or successive division by prime numbers. We start by dividing 24 by the smallest prime number, which is 2. We find that ${24 = 2 \times 12}$. Now, we need to factor 12. Again, we can divide 12 by 2, which gives us ${12 = 2 \times 6}$. Next, we factor 6, which is ${6 = 2 \times 3}$. Thus, we have broken down 24 into its prime factors: ${24 = 2 \times 2 \times 2 \times 3}$. This can be written more concisely as ${24 = 2^3 \times 3}$. The prime factorization of a number is unique, meaning that every number has only one set of prime factors. This property is crucial in many areas of mathematics, including number theory and cryptography. In our case, the prime factorization of 24 allows us to express ${\log_{10} 24}$ in terms of the logarithms of its prime factors, which are given in the problem statement. By understanding the prime factorization of 24, we can proceed to use logarithmic properties to simplify the expression and find the solution. This step is fundamental to solving the problem and demonstrates the importance of prime factorization in simplifying mathematical expressions.

Now that we have the prime factorization of 24 as ${2^3 \times 3}$, we can use logarithmic properties to express ${\log_{10} 24}$ in terms of ${\log_{10} 2}$ and ${\log_{10} 3}$. The key logarithmic properties we will use are the product 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, which can be written as ${\log_b(xy) = \log_b(x) + \log_b(y)}$. 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, which can be written as ${\log_b(x^p) = p \log_b(x)}$. Applying the product rule to ${\log_{10} 24}$, we have:

${ \log_{10} 24 = \log_{10} (2^3 \times 3) = \log_{10} (2^3) + \log_{10} 3 }$

Next, we apply the power rule to the term ${\log_{10} (2^3)}$:

${ \log_{10} (2^3) = 3 \log_{10} 2 }$

Substituting this back into the equation, we get:

${ \log_{10} 24 = 3 \log_{10} 2 + \log_{10} 3 }$

This expression now represents ${\log_{10} 24}$ in terms of ${\log_{10} 2}$ and ${\log_{10} 3}$, which are the given values ${m}$ and ${n}$, respectively. The correct application of these logarithmic properties is crucial in simplifying the expression and moving closer to the final solution. These rules are fundamental in working with logarithms and are used extensively in various mathematical and scientific applications. Understanding and applying these properties correctly allows us to transform complex logarithmic expressions into simpler forms, making calculations and problem-solving more manageable.

We have now expressed ${\log_{10} 24}$ in terms of ${\log_{10} 2}$ and ${\log_{10} 3}$ as:

${ \log_{10} 24 = 3 \log_{10} 2 + \log_{10} 3 }$

The problem statement provides us with the values ${\log_{10} 2 = m}$ and ${\log_{10} 3 = n}$. To find ${\log_{10} 24}$ in terms of ${m}$ and ${n}$, we simply substitute these values into the expression we derived:

${ \log_{10} 24 = 3(\log_{10} 2) + \log_{10} 3 = 3m + n }$

Therefore, ${\log_{10} 24}$ can be expressed as ${3m + n}$. This final substitution completes the solution, providing a concise expression for ${\log_{10} 24}$ in terms of the given variables ${m}$ and ${n}$. This step is straightforward but crucial, as it connects the logarithmic expression to the specific values provided in the problem. The ability to make such substitutions accurately is a fundamental skill in algebra and is essential for solving various mathematical problems. By substituting the given values, we have successfully transformed the logarithmic expression into a simple algebraic expression, demonstrating the power of logarithmic properties in simplifying mathematical problems. This solution not only answers the specific question but also illustrates a general method for expressing logarithms of composite numbers in terms of the logarithms of their prime factors.

In conclusion, given that ${\log_{10} 2 = m}$ and ${\log_{10} 3 = n}$, we have found that ${\log_{10} 24}$ can be expressed as ${3m + n}$. This result is obtained by first determining the prime factorization of 24 as ${2^3 \times 3}$, then applying the product and power rules of logarithms to express ${\log_{10} 24}$ in terms of ${\log_{10} 2}$ and ${\log_{10} 3}$, and finally substituting the given values of ${m}$ and ${n}$. This problem demonstrates the utility of logarithmic properties in simplifying complex expressions and relating the logarithms of different numbers. The ability to manipulate logarithmic expressions is a crucial skill in various fields, including mathematics, engineering, and computer science. By understanding and applying logarithmic properties, we can transform complex problems into simpler, more manageable forms. The solution presented here not only provides a specific answer to the problem but also illustrates a general method for solving similar problems involving logarithms. The logical progression from prime factorization to the application of logarithmic rules and the final substitution showcases a systematic approach to problem-solving in mathematics. This example reinforces the importance of understanding fundamental mathematical principles and their applications in solving practical problems.