Problem 42 - Intermediate Algebra

Question

Write each logarithm as a difference. Then simplify, if possible. See Example 3 . $$ \log _{8} \frac{y}{8} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\log_8 y - 1\).

Step 1: Recognize the Property of Logarithm

Identify the logarithm property that can help rewrite the expression. The property we need here is the quotient rule for logarithms: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). This property will allow us to express the logarithm of a fraction as a difference of logarithms.

Step 2: Apply the Quotient Rule

Apply the quotient rule to the given expression \( \log_8 \frac{y}{8} \). According to the rule, we can separate the numerator and the denominator: \( \log_8 y - \log_8 8 \).

Step 3: Simplify the Expression

Simplify the expression further. Since \( \log_8 8 = 1 \) (because 8 to the power of 1 is 8), substitute this into the expression to get \( \log_8 y - 1 \).

Key Concepts

Properties of Logarithms

Logarithms have specific properties that make them powerful tools for algebraic manipulation. One fundamental property that you often encounter is the ability to break down complicated expressions into simpler parts using logarithmic laws. These properties include:

• Product Rule: \( \log_b (MN) = \log_b M + \log_b N \)
• Quotient Rule: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)
• Power Rule: \( \log_b (M^p) = p \log_b M \)

These properties help in simplifying complex logarithmic expressions, which is crucial when solving equations that involve logarithms. For instance, when you encounter a logarithm of a fraction, you can use the quotient rule to split the fraction into a subtraction of two simpler logs.

Quotient Rule

When you see a logarithmic expression of a fraction, like \( \log_b \left( \frac{M}{N} \right) \), the quotient rule comes to the rescue. This rule tells us that we can break down the logarithm of a division into two separate logs that subtract from each other. In simpler terms, this rule says:

• Divide means subtract in log terms.
• The log of a numerator minus the log of a denominator.

Let's consider a real-life analogy. Imagine the original expression is a book that's too thick to read in one go. The quotient rule allows you to "split the book" into two smaller, more manageable books that you can tackle one at a time. This approach retains all the information but in a more digestible format. So, for the exercise \( \log_8 \frac{y}{8} \), applying the quotient rule rephrases it as \( \log_8 y - \log_8 8 \).

Simplification of Logs

The final step in handling logarithmic expressions often involves simplification. This step is key in making sense of the logarithmic terms you've extracted. For example, in the expression \( \log_8 y - \log_8 8 \), you need to realize that \( \log_8 8 = 1 \).

• Why? Because \( 8^1 = 8 \), which means the log of 8 with base 8 is 1.
• This realization lets you reduce the expression to \( \log_8 y - 1 \).

Understanding simplification helps strip away unnecessary parts of an expression until you're left with the most straightforward form. It's like taking layers off an onion—you get closer to the core idea with each simplification step. Mastering this skill ensures efficiency and clarity in solving logarithmic equations.