Problem 86 - Algebra 2

Question

Expand each logarithm. \(\log _{3}\left[(x y)^{\frac{1}{3}} \div z^{2}\right]^{3}\)

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \(\log_{3} (x y) - 2 \log_{3} z\).

Step 1: Apply power rule of logarithms

The power rule of logarithms states that \(\log_b(m^n) = n \cdot \log_b(m)\). Apply this rule to rewrite \(\log _{3}\left[(x y)^{\frac{1}{3}} \div z^{2}\right]^{3}\) as \(3 \log_{3}\left( (x y)^{\frac{1}{3}} \div z^{2}\right)\).

Step 2: Apply quotient rule of logarithms

The quotient rule of logarithms states that \(\log_b(m/n) = \log_b(m) - \log_b(n)\). Use this rule to rewrite \(3 \log_{3}\left( (x y)^{\frac{1}{3}} \div z^{2}\right)\) as \(3 \left(\log_{3} (x y)^{\frac{1}{3}} - \log_{3} z^{2} \right)\).

Step 3: Distribute the constant

Apply the distributive property to distribute the constant '3' to each term inside the parenthesis. The expression will then become \(3 \log_{3} (x y)^{\frac{1}{3}} - 3 \log_{3} z^{2}\).

Step 4: Apply power rule of logarithms again

Using the power rule of logarithms again, the expression \(3 \log_{3} (x y)^{\frac{1}{3}} - 3 \log_{3} z^{2}\) can be further expanded as \(\log_{3} (x y) - 2 \log_{3} z \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a handy tool when you encounter a logarithm with an exponent. It allows you to take the exponent and move it in front of the logarithm as a multiplier. This can simplify expressions and make it easier to understand what the logarithm represents.

In mathematical terms, the power rule is written as follows:

• \( \log_b(m^n) = n \cdot \log_b(m) \)

In our exercise with \( \log_{3}\left[(x y)^{\frac{1}{3}} \div z^{2}\right]^{3}\), we applied the power rule right at the start to handle the outer exponent of 3. We moved this 3 in front of the entire logarithmic expression, reducing the complexity of the problem.

This action transformed our expression into \( 3 \log_{3}\left( (x y)^{\frac{1}{3}} \div z^{2}\right) \). The power rule is a critical step when expanding logarithmic expressions with exponents and is often used multiple times in an exercise.

Quotient Rule of Logarithms

Often, in logarithms, you'll encounter divisions inside the argument. The quotient rule is your best friend in such a scenario. It helps break down a logarithm of a division into a difference of logarithms. The formula of the quotient rule is:

• \( \log_b(m/n) = \log_b(m) - \log_b(n) \)

In our problem, after using the power rule, we had:

• \( 3 \log_{3}\left( (x y)^{\frac{1}{3}} \div z^{2}\right) \)

By using the quotient rule, we expanded this to:

• \( 3 \left(\log_{3} (x y)^{\frac{1}{3}} - \log_{3} z^{2} \right) \)

The quotient rule is fundamental for splitting divided terms, making complex problems more manageable.

Distributive Property

The distributive property is an essential principle not just for algebra but also for expanding expressions involving logarithms. It allows us to multiply a number outside the parenthesis by each term within the parenthesis.

Consider the expression we handled after applying the quotient rule:

• \( 3 \left(\log_{3} (x y)^{\frac{1}{3}} - \log_{3} z^{2} \right) \)

Here, the number 3 needs to be distributed across the terms inside the parenthesis. So, we apply the distributive property:

• \( 3 \log_{3} (x y)^{\frac{1}{3}} - 3 \log_{3} z^{2} \)

This step simplifies each term individually, allowing further application of logarithmic properties.

Logarithmic Properties

Logarithmic properties are a set of rules that help in transforming and simplifying logarithmic expressions. Two of the most commonly used properties are the power and quotient rules, but there are more aspects to consider.

After expanding our expression using the distributive property, we used the power rule again to deal with following expressions:

• \( 3 \log_{3} (x y)^{\frac{1}{3}} \)
• \( 3 \log_{3} z^{2} \)

Applying the power rule to each of these terms, we got:

• \( \log_{3} (x y) \)
• \( - 2 \log_{3} z \)

These transformations highlight how logarithmic properties allow us to rearrange and transform expressions in ways that reveal deeper insights into their structure.

Learning and using these properties well can simplify complex algebraic problems and boost problem-solving efficiency in mathematics.