Problem 58 - College Algebra and Calculus: An Applied Approach

Question

Use the properties of logarithms to simplify the given logarithmic expression.\(\log _{5} \frac{1}{15}\)

Step-by-Step Solution

Answer

The simplified logarithmic expression is \(-2 - \log _{5} 3\).

Step 1: Apply the Quotient Rule

The quotient rule states that \(\log _{b}\frac{a}{c} = \log _{b}a - \log _{b}c\). Apply this rule to the provided expression \(\log _{5} \frac{1}{15}\), which gives us \(\log _{5} 1 - \log _{5} 15\).

Step 2: Simplify the Logarithms

Any logarithm with 1 as input results in 0, regardless of the base. Hence, \(\log _{5} 1 = 0\). Therefore, the expression is simplified to \(0 - \log _{5} 15\). We also know that \(\log _{5} 5 = 1\). Hence, \(\log _{5} 15 = \log _{5} 5^2 + \log _{5} 3 = 2 + \log _{5} 3\). Hence the expression is further simplified to \(0 - (2 + \log _{5} 3)\).

Step 3: Simplify the Expression

Apply the properties of the real numbers to gather the constants and logarithms separately. The expression simplifies to \(-2 - \log _{5} 3\)

Key Concepts

Properties of Logarithms

When it comes to logarithms, certain properties can greatly simplify calculations and expressions. These properties include:

• Product Rule: \ \( \log_b(xy) = \log_b x + \log_b y \)
• Quotient Rule: \ \( \log_b(\frac{x}{y}) = \log_b x - \log_b y \)
• Power Rule: \ \( \log_b(x^p) = p \log_b x \)

These rules are powerful because they allow us to transform complex logarithmic expressions into simpler forms. This is helpful when you want to combine or break apart logarithmic terms. In our exercise, we used the Quotient Rule to break down \ \( \log_5 \frac{1}{15} \) into smaller, more manageable pieces.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra and calculus, as it allows you to reduce complex equations into simpler forms. This process often involves using the properties of the operations involved, such as addition, subtraction, multiplication, and division.
When dealing with logarithmic expressions, applying the properties of logarithms can simplify the calculations. For instance, in our example, by recognizing that \ \( \log_5 1 = 0 \), we were able to simplify \ \( \log_5 \frac{1}{15} \) to \ \( 0 - \log_5 15 \).
The key is to identify and apply the appropriate property that will simplify the expression without changing its value. This process not only makes calculations easier but also helps in understanding the underlying structure of the mathematical problem at hand.

Quotient Rule for Logarithms

When you encounter a logarithmic expression in the form \ \( \log_b \frac{a}{c} \), the quotient rule is a handy tool to simplify it. This rule states that \ \( \log_b \frac{a}{c} = \log_b a - \log_b c \). By splitting the fraction into a subtraction of two logarithmic terms, you can work with simpler expressions
In practical terms, the quotient rule breaks a complex fraction into two easier parts. For example, in the exercise, \ \( \log_5 \frac{1}{15} = \log_5 1 - \log_5 15 \). Here, knowing that \ \( \log_5 1 \) equals zero simplifies the expression significantly.
This rule showcases the importance of understanding mathematical properties and how they assist in reducing calculation efforts, especially when handling intricate logarithmic problems.