Problem 44 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

\( \log_{5}(x) - 4 \cdot \log_{5}(y) \)

Step 1: Apply the Quotient Rule for Logarithms

The expression \( \log_{5} \left( \frac{x}{y^4} \right) \) can be rewritten using the quotient rule for logarithms, which states that \( \log_{b}(\frac{M}{N}) = \log_{b}(M) - \log_{b}(N) \). Thus, we have: \[ \log_{5} \left( \frac{x}{y^4} \right) = \log_{5}(x) - \log_{5}(y^4) \]

Step 2: Apply the Power Rule for Logarithms

The term \( \log_{5}(y^4) \) can be further simplified using the power rule, which states that \( \log_{b}(M^n) = n \cdot \log_{b}(M) \). Applying this rule, we get: \[ \log_{5}(y^4) = 4 \cdot \log_{5}(y) \]

Step 3: Write the Expression as a Sum or Difference of Logarithms

Substituting the expression for \( \log_{5}(y^4) \) back into the equation from Step 1, the expression becomes: \[ \log_{5}(x) - \log_{5}(y^4) = \log_{5}(x) - 4 \cdot \log_{5}(y) \] Thus, the expression \( \log_{5} \left( \frac{x}{y^4} \right) \) is written as a difference of logarithms: \[ \log_{5}(x) - 4 \cdot \log_{5}(y) \]

Key Concepts

Understanding the Quotient Rule

The quotient rule for logarithms is a valuable tool to simplify logarithmic expressions, especially those involving division inside the log. When you encounter a log of a fraction, such as \( \log_{b}(\frac{M}{N}) \), the quotient rule allows you to break it down into two separate logs: \( \log_{b}(M) \) and \( \log_{b}(N) \). The rule states that the log of a ratio is equal to the difference of the logs. This means you subtract the log of the denominator (bottom part) from the log of the numerator (top part).

Here's how it works in simple terms:

• If you have \( \log_{5}(\frac{x}{y^4}) \), the quotient rule transforms it into \( \log_{5}(x) - \log_{5}(y^4) \).
• By separating it in this way, you make it easier to apply further rules, like the power rule, to simplify the expression even more.

This separation step is crucial for further simplification and makes manipulation of logarithmic expressions more manageable.

Mastering the Power Rule

The power rule for logarithms is another essential tool that helps simplify expressions by dealing with exponents inside the log. It states that \( \log_{b}(M^n) = n \cdot \log_{b}(M) \). This means that you can "move" the exponent out of the log as a coefficient.

Let's dive into how it applies:

• In the expression \( \log_{5}(y^4) \), the exponent 4 can be brought in front of the log, turning it into \( 4 \cdot \log_{5}(y) \).
• This adjustment allows you to express the term more simply and makes the entire expression easier to handle.

By using the power rule, you reduce the complexity of the logarithmic expression, often making it possible to combine or compare it with other terms.

Simplifying Expressions Using Logarithms

Simplifying logarithmic expressions is all about applying the right rules at the right time to make the expression more user-friendly and less chaotic. The process typically involves using both the quotient and power rules:

• First, identify parts of the expression where division occurs and use the quotient rule to split these into simpler parts.
• Next, look for powers (or exponents) in these parts and apply the power rule, moving the exponent outside the log.
• Finally, combine these parts into a cohesive, simplified expression.

For example, starting with \( \log_{5}(\frac{x}{y^4}) \), splitting into \( \log_{5}(x) - \log_{5}(y^4) \) using the quotient rule, and then simplifying to \( \log_{5}(x) - 4 \cdot \log_{5}(y) \) using the power rule gives you a much more streamlined expression. These steps untangle the expression, revealing its fundamental components in a more digestible form.