Problem 43 - 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 \(y\) $$ \log \frac{1}{5} $$

Step-by-Step Solution

Answer

\( \log \frac{1}{5} = -y \).

Step 1: Identify the Given

We have the expression \( \log \frac{1}{5} \), and we are given that \( \log 3 = x \) and \( \log 5 = y \). We need to express \( \log \frac{1}{5} \) in terms of \( x \) and \( y \).

Step 2: Apply Logarithmic Identity

Recall the logarithmic identity for the reciprocal of a number: \( \log \left( \frac{1}{a} \right) = -\log a \). Here, \( a = 5 \).

Step 3: Substitute and Simplify

Using the identity from Step 2, substitute \( a = 5 \) into the equation: \( \log \frac{1}{5} = -\log 5 \). Since \( \log 5 = y \), we can write \( \log \frac{1}{5} = -y \).

Key Concepts

Logarithmic Identities

Understanding logarithmic identities is fundamental in simplifying logarithmic expressions. One important identity is for the reciprocal of a number, which states:

• \( \log \left( \frac{1}{a} \right) = -\log a \)

This means that the logarithm of the reciprocal of a number is equal to the negative of the logarithm of the number itself.
In the original exercise, we apply this identity by recognizing that \( \frac{1}{5} \) is the reciprocal of 5. Thus, \( \log \frac{1}{5} \) becomes \(-\log 5\). This simple yet powerful identity allows us to write complicated logarithms in terms of simpler ones. It's a handy tool for mathematical manipulation and simplification.
By using these identities, you can easily transform and work with logarithmic expressions in various mathematical problems.

Change of Base

The change of base formula is a crucial tool for evaluating and converting logarithms. It is particularly useful when a specific base is not suitable for direct calculation. The formula is expressed as:

• \( \log_b a = \frac{\log_k a}{\log_k b} \)

where \(k\) is any positive number that provides a convenient base, often 10 or \(e\) (Euler's number) for practical calculations.
Although the original exercise does not directly involve changing the base, understanding this concept can help when you need to calculate logarithms in different bases or when you prefer a base that matches the information given.
For instance, if you encounter \( \log_3 5 \) and would like to express it using base 10, you could use the change of base formula to rewrite it as \( \frac{\log_{10} 5}{\log_{10} 3} \). This technique simplifies complex expressions and aids in numerical calculations with pattern recognition and technological tools.

Properties of Logarithms

Properties of logarithms are key to handling logarithmic expressions efficiently. These properties help break down complex logarithmic expressions into basic components. Here are the core properties:

• Product Property: \( \log_b (mn) = \log_b m + \log_b n \)
• Quotient Property: \( \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n \)
• Power Property: \( \log_b (m^n) = n \cdot \log_b m \)

Each of these properties simplifies expressions by transforming them into sums, differences, or products, which are easier to evaluate or convert.
In the original exercise, our focus was using a specific identity, but these properties are the foundation that makes such transformations possible and help solve a wide range of problems.
For example, if you have the expression \( \log (15) \) and you know \( \log 3 = x \) and \( \log 5 = y \), you can express it as \( \log (3 \times 5) = \log 3 + \log 5 = x + y \). This makes tackling logarithmic problems more straightforward and manageable.