Problem 39 - 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 \(y\) $$ \log 45 $$

Step-by-Step Solution

Answer

\( \log 45 = 2x + y \)

Step 1: Understanding the Given

We are provided with the values of two logarithms: \( \log 3 = x \) and \( \log 5 = y \). The exercise requires expressing the logarithm \( \log 45 \) in terms of \( x \) and \( y \).

Step 2: Express 45 in Terms of Known Bases

Recognize that 45 can be expressed in terms of the factors 3 and 5. Specifically, \( 45 = 3^2 \cdot 5 \).

Step 3: Use Logarithm Properties

Apply the logarithm of a product property: \( \log(a \cdot b) = \log a + \log b \). Substitute the factorization from the previous step: \( \log(45) = \log(3^2 \cdot 5) = \log(3^2) + \log 5 \).

Step 4: Apply the Power Rule

Use the power rule of logarithms: \( \log(a^b) = b \cdot \log a \). Apply this to \( \log(3^2) \) to get \( 2 \cdot \log 3 \).

Step 5: Substitute Known Values

Substitute the known values for \( \log 3 \) and \( \log 5 \) from the given: \( \log(3^2) = 2x \) and \( \log 5 = y \). Therefore, \( \log(45) = 2x + y \).

Step 6: Final Expression

Combine the results to express \( \log 45 \) in terms of \( x \) and \( y \): \( \log 45 = 2x + y \).

Key Concepts

Logarithm of a Product

The logarithm of a product is a handy property of logarithms that simplifies our calculations. It states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers. This can be written as follows:

• \(\log(a \cdot b) = \log a + \log b\)

Imagine you have numbers \(a\) and \(b\) and wish to find the logarithm of their product, \(a \cdot b\). Instead of dealing with a possibly complex multiplication inside a logarithm, you can break it down into simpler parts.
This property comes in handy when you're working with numbers like 45, as was demonstrated in the original exercise. By expressing 45 as the product of its prime factors, specifically 3 and 5 squared, you can use this property to simplify the logarithm problem into more manageable parts.
In practice, if you know \(\log 3=x\) and \(\log 5=y\), you combine these values using the property above to find \(\log 45\) by recognizing \(45 = 3^2 \cdot 5\). Applying \(\log(45) = \log(3^2 \cdot 5) = \log(3^2) + \log 5\).

Power Rule of Logarithms

The power rule of logarithms is a powerful tool that allows you to take an exponent and bring it in front of the logarithm. This is particularly useful when dealing with logarithms of exponential numbers. The rule states:

• \(\log(a^b) = b \cdot \log a\)

This means if you have an exponent inside a log function, you can "move" it to the front, turning multiplication inside the logarithm into multiplication outside.
This simplifies calculations significantly, especially when dealing with compounds like \(3^2\) in our example. Instead of computing \(\log(3^2)\) directly, we use the power rule: \(\log(3^2) = 2 \cdot \log 3\). Now, if you know \(\log 3 = x\), then \(\log(3^2) = 2x\).
Using this rule effectively can transform complex logarithmic expressions into simpler and more manageable forms, which is essential in both algebra and calculus.

Expressing Logarithms in Terms of Variables

Expressing logarithms in terms of variables is a common task in algebra, especially when you need to relate known values to new expressions. This involves taking known log values and using algebraic techniques to express other logs in terms of these known quantities.
In the example exercise, we know two specific logarithm values: \(\log 3 = x\) and \(\log 5 = y\). Using these, the challenge is to express \(\log 45\) in terms of \(x\) and \(y\).

• Recognize that \(45 = 3^2 \cdot 5\) allows us to break it into known parts using logarithmic properties.
• Apply the product rule: \(\log(3^2 \cdot 5) = \log(3^2) + \log 5\).
• Use the power rule to further break down: \(\log(3^2) = 2 \cdot \log 3 = 2x\).

Finally, combining all these, we express \(\log 45\) as \(2x + y\), neatly tying the sought logarithm to the known variables. This approach is invaluable in simplifying complex expressions, allowing easier computation and interpretation of logarithmic relationships.