Expressing Log 24 In Terms Of Log 2 And Log 3 A Comprehensive Guide

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Understanding logarithms is crucial in various fields, from mathematics and physics to computer science and finance. One of the fundamental properties of logarithms is their ability to simplify complex calculations by breaking down numbers into their prime factors. This article will delve into expressing the logarithm of 24 in terms of the logarithms of 2 and 3, providing a step-by-step explanation and highlighting the underlying logarithmic principles.

Understanding Logarithms

Before we dive into the specifics, let's briefly recap what logarithms are. A logarithm answers the question: "To what power must we raise a base number to get a certain value?" In mathematical terms, if b^x = y, then the logarithm of y to the base b is x, written as log_b(y) = x. When no base is explicitly written (e.g., log(y)), it is generally assumed to be base 10, known as the common logarithm. Natural logarithms use the base e (Euler's number, approximately 2.71828) and are written as ln(y).

Logarithms possess several key properties that make them incredibly useful:

  1. Product Rule: log_b(mn) = log_b(m) + log_b(n)
  2. Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
  3. Power Rule: log_b(m^k) = k * log_b(m)
  4. Change of Base Rule: log_b(a) = log_c(a) / log_c(b)

These properties allow us to manipulate logarithmic expressions, making it easier to simplify and solve equations. In our case, we'll primarily use the product rule and the power rule to express log 24 in terms of log 2 and log 3.

Prime Factorization of 24

The first step in expressing log 24 in terms of log 2 and log 3 is to find the prime factorization of 24. Prime factorization is the process of breaking down a number into its prime factors, which are the prime numbers that multiply together to give the original number. A prime number is a number greater than 1 that has only two divisors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, and so on.

To find the prime factorization of 24, we can start by dividing it by the smallest prime number, 2:

  • 24 ÷ 2 = 12
  • 12 ÷ 2 = 6
  • 6 ÷ 2 = 3

Now we have reached 3, which is also a prime number. Thus, the prime factorization of 24 is 2 × 2 × 2 × 3, which can be written as 2^3 × 3. Understanding this prime factorization is the key to expressing log 24 in the desired form.

Expressing log 24 in Terms of log 2 and log 3

Now that we have the prime factorization of 24 (2^3 × 3), we can use the properties of logarithms to express log 24 in terms of log 2 and log 3. Recall the product rule, which states that log_b(mn) = log_b(m) + log_b(n). Applying this rule to log 24, we get:

log 24 = log (2^3 × 3)

Using the product rule, we can separate the logarithm of the product into the sum of the logarithms:

log (2^3 × 3) = log (2^3) + log (3)

Next, we'll use the power rule, which states that log_b(m^k) = k * log_b(m). Applying this rule to log (2^3), we get:

log (2^3) = 3 * log 2

Now we can substitute this back into our equation:

log 24 = 3 * log 2 + log 3

Thus, log 24 can be expressed as 3 log 2 + log 3. This matches option C in the original question. This expression shows how the logarithm of a composite number like 24 can be broken down into the logarithms of its prime factors, 2 and 3, making it easier to work with in various mathematical contexts.

Why This Matters: Applications and Implications

Expressing logarithms in terms of their prime factors isn't just a mathematical exercise; it has practical applications in various fields:

  1. Simplifying Calculations: Logarithms transform multiplication into addition, which can significantly simplify complex calculations. By expressing a logarithm in terms of its prime factors, we can break down a complex logarithm into simpler components.
  2. Solving Exponential Equations: Logarithms are essential for solving exponential equations. When an unknown variable is in the exponent, logarithms provide a way to bring the exponent down as a coefficient, making the equation solvable.
  3. Scales and Measurement: Logarithmic scales are used in various fields to represent large ranges of values in a more manageable way. Examples include the Richter scale for earthquake magnitude, the decibel scale for sound intensity, and the pH scale for acidity.
  4. Computer Science: Logarithms are fundamental in computer science, particularly in analyzing the efficiency of algorithms. The time complexity of many algorithms is expressed using logarithmic notation (e.g., O(log n)), which indicates how the runtime of the algorithm grows with the input size.
  5. Finance: Logarithms are used in finance to calculate compound interest, depreciation, and other financial metrics. Understanding logarithmic relationships is crucial for financial modeling and analysis.

In summary, expressing log 24 in terms of log 2 and log 3 is a fundamental exercise that highlights the power and utility of logarithmic properties. This skill is applicable in various contexts, making it an essential concept for anyone studying mathematics, science, or related fields.

Common Mistakes and How to Avoid Them

When working with logarithms, it's easy to make mistakes if you're not careful. Here are some common errors and how to avoid them:

  1. Incorrectly Applying Logarithmic Properties:

    • Mistake: Applying the product rule as log(m + n) = log(m) + log(n). This is incorrect; the correct rule is log(mn) = log(m) + log(n).
    • How to Avoid: Always double-check the logarithmic properties before applying them. Remember the product rule, quotient rule, and power rule, and use them correctly.

  2. Mixing Up Bases:

    • Mistake: Forgetting to specify the base of the logarithm or mixing up different bases in the same equation.
    • How to Avoid: Always be clear about the base of the logarithm. If no base is written, it's usually assumed to be base 10. If you're working with natural logarithms, make sure to use the ln notation.

  3. Misunderstanding the Power Rule:

    • Mistake: Applying the power rule incorrectly, such as log(m^k) = (log m)^k.
    • How to Avoid: Remember that the power rule states log_b(m^k) = k * log_b(m). The exponent k applies only to the argument m, not to the entire logarithm.

  4. Forgetting the Domain of Logarithms:

    • Mistake: Taking the logarithm of a non-positive number (zero or a negative number).
    • How to Avoid: Logarithms are only defined for positive arguments. Always ensure that the argument of the logarithm is greater than zero.

  5. Incorrect Prime Factorization:

    • Mistake: Incorrectly factoring a number into its prime factors.
    • How to Avoid: Double-check your prime factorization to ensure that you have broken down the number correctly. Use a systematic approach, such as dividing by the smallest prime numbers first.

  6. Confusing Logarithmic and Exponential Forms:

    • Mistake: Mixing up the logarithmic form (log_b(y) = x) with the exponential form (b^x = y).
    • How to Avoid: Practice converting between logarithmic and exponential forms. Understanding the relationship between these forms is crucial for solving logarithmic equations.

By being aware of these common mistakes and practicing regularly, you can improve your understanding of logarithms and avoid errors in your calculations.

Conclusion

Expressing log 24 in terms of log 2 and log 3 demonstrates the power and flexibility of logarithmic properties. By understanding the prime factorization of a number and applying the product and power rules of logarithms, we can simplify complex expressions and gain deeper insights into mathematical relationships. This skill is not only valuable in academic settings but also has practical applications in various fields, highlighting the importance of mastering logarithmic concepts. Remember to always double-check your work, apply the properties correctly, and be mindful of common mistakes to ensure accurate and efficient problem-solving.