Problem 40 - Algebra 2 and Trigonometry

Question

\(\operatorname{In} 36-47,\) if \(\log 3=x\) and \(\log 5=y,\) write each of the logs in terms of \(x\) and \(y\) $$ \log 75 $$

Step-by-Step Solution

Answer

\( \log 75 = x + 2y. \)

Step 1: Prime Factorization of 75

Start by expressing the number 75 as a product of its prime factors. We know that 75 is equal to 3 times 25, and 25 can be further broken down into 5 times 5. Therefore, the prime factorization of 75 is \( 75 = 3 \times 5^2 \).

Step 2: Expressing Logarithm Using Prime Factors

Now, express \( \log 75 \) in terms of its prime factors. Using the property of logarithms \( \log(ab) = \log a + \log b \), we find \[ \log 75 = \log(3 \times 5^2) = \log 3 + \log(5^2) = \log 3 + 2\log 5. \]

Step 3: Substitute Given Values

Substitute the given values of \( \log 3 = x \) and \( \log 5 = y \) into the equation from Step 2. Replacing these values, we have:\[ \log 75 = x + 2y. \]

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime numbers. A prime number is a whole number greater than 1 that can only be divided by 1 and itself without leaving a remainder. For the number 75, we can use prime factorization to break it down into its basic building blocks. We start by dividing 75 by the smallest prime number, which is 3. This gives us 25. Then, since 25 itself is not a prime number, we further divide it by 5, resulting in 5, which is a prime number. Thus, the prime factorization of 75 is:

• 3, because 75 divided by 3 equals 25, and 3 is a prime number.
• 5 squared (\(5^2\)), because 25 is obtained by multiplying 5 by itself.

Hence, we can represent 75 as \(75 = 3 \times 5^2\). Understanding prime factorization is crucial for simplifying expressions in mathematics, especially when working with logarithms.

Logarithmic Properties

Logarithmic properties are rules that simplify the manipulation and calculation of logarithms. These properties are essential when dealing with expressions involving logarithms, especially as seen in the study of logarithmic transformations. The key properties often used include:

• Product Rule: \(\log(ab) = \log a + \log b\), which helps break down the logarithm of a product into a sum of separate logarithms.
• Power Rule: \(\log(a^b) = b \cdot \log a\), which is useful when a logarithm has an exponent and allows us to move the exponent in front as a coefficient.

Using these properties, the expression \(\log 75\) can be rewritten using its prime factorization from the earlier section as \(\log(3 \times 5^2)\). This becomes \(\log 3 + \log(5^2)\), which simplifies to \(\log 3 + 2\log 5\). Understanding these properties is essential for simplifying complex logarithmic expressions in mathematical problems.

Expression Transformation

Expression transformation involves rewriting a mathematical expression in a different form, usually to simplify computation or to reveal certain relationships. In the context of logarithms, it often involves using known values to substitute into more complex expressions. In this exercise, you are tasked with expressing \(\log 75\) in terms of \(x\) and \(y\), where \(\log 3 = x\) and \(\log 5 = y\).After using prime factorization and logarithmic properties, you have the equation \(\log 75 = \log 3 + 2\log 5\). By substituting the given values, it becomes clearer:

• Replace \(\log 3\) with \(x\).
• Replace \(\log 5\) with \(y\).

This transformation results in the expression \(\log 75 = x + 2y\). This final form is simpler and expresses \(\log 75\) entirely in terms of the known variables \(x\) and \(y\). Mastering expression transformation is crucial for solving complex algebraic and logarithmic problems efficiently.