Problem 80 - Algebra 2

Question

Expand each logarithm. \(\log \frac{m^{3}}{n^{4} p^{-2}}\)

Step-by-Step Solution

Answer

The simplified form of the given logarithmic expression \(\log \frac{m^{3}}{n^{4} p^{-2}}\) is \( 3 \log m - 4 \log n + 2 \log p \).

Step 1: Apply the quotient rule

First, apply the quotient rule which states that \(\log \frac{m}{n} = \log m - \log n\), this will transform \(\log \frac{m^{3}}{n^{4} p^{-2}}\) to \( \log m^{3} - \log(n^{4} p^{-2})\).

Step 2: Apply both the quotient and product rules

Following that, expand the negative logarithm term by applying the quotient rule again to \(\log(n^{4}p^{-2})\), which results in \( \log m^{3} - \log n^{4} + \log p^{2}\). It's positive for \( \log p^{2}\) because we're subtracting a negative log. The product rule was also applied since the original expression was in the numerator and denominator form (respectively).

Step 3: Apply the power rule

Lastly, apply the power rule, which states that \(\log m^n = n \log m\), to each term. The power rule will transform each term into \( 3 \log m - 4 \log n + 2 \log p \).

Key Concepts

Quotient Rule

The quotient rule is an essential property of logarithms that simplifies the log of a division into the difference of logs. It's expressed as \( \log \left( \frac{a}{b} \right) = \log a - \log b \).
This rule helps when you have a fraction within a logarithm, allowing you to separate it into two distinct terms. For instance, in the original problem \( \log \frac{m^3}{n^4 p^{-2}} \), the quotient rule begins this expansion by changing it into \( \log m^3 - \log(n^4 p^{-2}) \).

This step separates the term into a more manageable format, making it easier to apply other logarithm rules. Remember, this is like turning a subtraction outside a logarithm into a division inside one. It comes in handy when simplifying complex log expressions by breaking down the terms.

Power Rule

The power rule allows you to move an exponent from inside a logarithm to outside as a coefficient. It's defined as \( \log a^b = b \log a \).
This rule is particularly useful because it lets you handle logarithms with exponents more efficiently.

In the exercise, after applying the quotient and product rules, you encounter terms like \( \log m^3 \), \( \log n^4 \), and \( \log p^2 \). Using the power rule, these terms transform into:

• \( 3 \log m \)
• \(- 4 \log n \)
• \(2 \log p \)

This makes calculations easier and often cleaner, converting exponentiation inside a log to simple multiplication outside. It's a handy tool when expanding logarithms with variables raised to powers.

Product Rule

The product rule for logarithms allows you to split a log containing a product into a sum of individual logs. The rule is represented as \( \log(ab) = \log a + \log b \).

In our exercise, this rule helps refine the expanded form once you apply the quotient rule again. When dealing with \( \log(n^4 p^{-2}) \), you can express it as \( \log n^4 + \log p^{-2} \). However, remember since we're subtracting this section, we need to adjust signs, resulting in \( -\log n^4 \) and \(+ \log p^2 \), due to the subtraction of a negative, which flips the sign.

This step simplifies complex nested logs, breaking them into parts that use other rules more straightforwardly. It's necessary to make sure each component log is as basic as it can be.