Problem 39 - Algebra A Combined Function

Question

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{3} \frac{4 y}{5} $$

Step-by-Step Solution

Answer

\( \log_3 4 + \log_3 y - \log_3 5 \).

Step 1: Apply the Quotient Rule of Logarithms

The quotient rule of logarithms states that \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). Apply this to the given expression:\[ \log_3 \frac{4y}{5} = \log_3 4y - \log_3 5. \]

Step 2: Apply the Product Rule of Logarithms

The product rule of logarithms states that \( \log_b(MN) = \log_b M + \log_b N \). Apply this to \( \log_3 4y \):\[ \log_3 4y = \log_3 4 + \log_3 y. \]

Step 3: Combine the Results

Substitute the result from step 2 into the result from step 1 to get the final expression as a sum and difference of logarithms:\[ \log_3 \frac{4y}{5} = \log_3 4 + \log_3 y - \log_3 5. \]

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a handy tool for simplifying expressions where a division is present inside a logarithm. This rule states that for any positive numbers \( M \) and \( N \), and a logarithm base \( b \), the logarithmic expression can be rewritten as follows:\[\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N.\]This rule makes it possible to split a single logarithm into a difference of two separate logarithms. The key motivation for using this rule is to simplify the expression and make it easier to work with. For example, in the original problem, we have \( \log_3 \frac{4y}{5} \). By applying the quotient rule, this becomes \( \log_3 4y - \log_3 5 \). Notice how we separate the numerator \( 4y \) and the denominator \( 5 \) as two distinct log terms. With this step, we've managed to transform a more complex expression into something simpler to analyze.

Product Rule of Logarithms

The product rule of logarithms is used to deal with multiplication inside a logarithm. This rule states that for positive numbers \( M \) and \( N \), with a logarithm base \( b \), the logarithm of a product can be expressed as a sum:\[\log_b (MN) = \log_b M + \log_b N.\]In other words, when you see multiplication within a logarithmic expression, you can "split" it into the sum of two separate logarithms. This makes dealing with logarithmic expressions significantly easier, particularly when solving equations or simplifying expressions.Returning to the problem, after we've isolated \( \log_3 4y \) using the quotient rule, we can now apply the product rule. The expression \( \log_3 4y \) can be expanded to \( \log_3 4 + \log_3 y \). This helps in breaking down the expression into simpler parts, each corresponding to a single term, allowing you to tackle them individually.

Logarithm Properties

Understanding logarithm properties is crucial to manipulating and simplifying logarithmic expressions. These properties guide us in converting complex expressions into simpler forms, making them easier to handle.Some of the fundamental properties include:

• Power Rule: \( \log_b (M^p) = p \cdot \log_b M \), which allows exponents to be brought down as a coefficient.
• Quotient Rule: As discussed, \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \), which separates division into a difference.
• Product Rule: As explained, \( \log_b (MN) = \log_b M + \log_b N \), converting multiplication to addition.

Knowing these rules allows us to dissect and reconstruct expressions with ease. In the original exercise, these properties enabled us to change \( \log_3 \frac{4y}{5} \) into \( \log_3 4 + \log_3 y - \log_3 5 \), facilitating a clearer understanding and making potential future calculations much more straightforward. They are the building blocks for working with any logarithmic expressions effectively.