Problem 16 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to expand the expression. $$ \log _{5} \frac{x}{2} $$

Step-by-Step Solution

Answer

\( \log_{5} x - \log_{5} 2 \)

Step 1: Recognize the Logarithmic Property

Identify the logarithmic property needed for expansion. Here, we use the Quotient Rule for logarithms, which states that \( \log_b \frac{m}{n} = \log_b m - \log_b n \).

Step 2: Apply the Quotient Rule

Apply the quotient rule to the expression \( \log_{5} \frac{x}{2} \). This gives us: \( \log_{5} x - \log_{5} 2 \).

Step 3: Simplify Final Expression

The expression \( \log_{5} x - \log_{5} 2 \) is already as simplified as possible using the laws of logarithms for this context.

Key Concepts

Quotient Rule

The Quotient Rule is one of the fundamental properties in the study of logarithms. It comes into play when you are dealing with a logarithm of a division, like \( \log_b \frac{m}{n} \). The rule indicates that the logarithm of a quotient is equivalent to the difference between the logarithms of the numerator and the denominator. So in mathematical terms, \( \log_b \frac{m}{n} = \log_b m - \log_b n \). This property simplifies calculations by transforming division into subtraction, which is often more straightforward to work with.

• It's not just a practical tool for simplification; it's vital for expanding expressions involving logarithms.
• It aligns with the basic principles of logarithms, where addition aligns with multiplication and subtraction aligns with division.

So in our original problem, when you see \( \log_{5} \frac{x}{2} \), think about it as \( \log_{5} x - \log_{5} 2 \). That’s the Quotient Rule in action.

Logarithmic Properties

Logarithmic properties are essential rules that help you manipulate and simplify expressions involving logarithms. They are based on the idea that logarithms are the inverses of exponential functions. Here's a quick reminder of some basic properties:

• The Product Rule: \( \log_b (mn) = \log_b m + \log_b n \)
• The Power Rule: \( \log_b (m^n) = n \cdot \log_b m \)
• The Quotient Rule: This was covered in detail above, \( \log_b \frac{m}{n} = \log_b m - \log_b n \).

Understanding these properties allows you to expand, condense, and manipulate logarithmic expressions effectively. For instance, in the original exercise, using the Quotient Rule helped us break down \( \log_{5} \frac{x}{2} \) into more manageable parts. By knowing these properties, you can handle other logarithmic expressions just as easily.

Logarithmic Expansion

Logarithmic expansion involves breaking down a more complex logarithmic expression into simpler parts to make it easier to work with or understand. This process utilizes the properties of logarithms, especially the ones like the Quotient Rule, Product Rule, and Power Rule. When you expand a logarithmic expression, you are essentially transforming the expression into a sum or difference of simpler logarithms.
The original exercise was a great example. We used logarithmic expansion to transform \( \log_{5} \frac{x}{2} \) into \( \log_{5} x - \log_{5} 2 \).
This is beneficial because:

• It allows for easier computation, especially with more complicated numbers.
• It provides a clear visual representation of the relationship between the components of the logarithm.
• It aids in identifying properties that can simplify solving equations.

Overall, logarithmic expansion is a simple yet powerful technique that builds a bridge between complex expressions and easy-to-handle components.