Problem 265 - Precalculus

Question

For the following exercise, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as asum, difference, or product of logs. $$ \log \left(\sqrt{x^{3} y^{-4}}\right) $$

Step-by-Step Solution

Answer

\( \frac{3}{2}\log(x) - 2\log(y) \)

Step 1: Rewrite the Radical

Begin by expressing the square root as an exponent. The expression under the logarithm becomes \( x^{3} y^{-4} \) raised to the power of \( \frac{1}{2} \). This gives us: \( \log((x^3 y^{-4})^{1/2}) \).

Step 2: Apply the Power Rule

Use the power rule for logarithms, which states \( \log(a^b) = b \cdot \log(a) \). Apply this rule to get: \( \frac{1}{2} \cdot \log(x^3 y^{-4}) \).

Step 3: Use the Product Rule

The expression inside the logarithm is a product, \( x^3 y^{-4} \), so apply the product rule: \( \log(ab) = \log(a) + \log(b) \). This gives you: \( \frac{1}{2} (\log(x^3) + \log(y^{-4})) \).

Step 4: Use the Power Rule Again

Apply the power rule individually to each logarithm: \( \log(x^3) = 3\log(x) \) and \( \log(y^{-4}) = -4 \log(y) \). After applying, the expression becomes \( \frac{1}{2} (3\log(x) - 4\log(y)) \).

Step 5: Simplify the Expression

Distribute the \( \frac{1}{2} \) to both terms to simplify: \( \frac{3}{2}\log(x) - 2\log(y) \). This is the expanded form of the original logarithm.

Key Concepts

Power Rule for Logarithms

To understand how to expand logarithmic expressions effectively, it's important to grasp the Power Rule for Logarithms. This rule is extremely handy when dealing with log expressions involving exponents. It states that if you have a logarithm of a number raised to an exponent, you can bring the exponent out in front of the log. Here's what it looks like:

• If you have \( \log(a^b) \), then it simplifies to \( b \cdot \log(a) \).

In our problem, we encountered \( (x^3 y^{-4})^{1/2} \). This entire expression is raised to the power of \( \frac{1}{2} \), so applying the Power Rule allowed us to take \( \frac{1}{2} \) outside the logarithm, simplifying the expression to \( \frac{1}{2} \cdot \log(x^3 y^{-4}) \).That is a crucial first step to break down complex log expressions!Remember, any exponent inside the log can be moved outside to make calculations easier.

Product Rule for Logarithms

The Product Rule for Logarithms comes into play when you have a logarithm of a product, meaning two terms multiplied together inside the log. This rule allows you to split the log into the sum of two separate logs. It's formulated as:

• If you have a situation like \( \log(ab) \), you can rewrite it as \( \log(a) + \log(b) \).

In our example, after using the Power Rule, we encountered \( \log(x^3 y^{-4}) \). By applying the Product Rule, this was transformed into \( \log(x^3) + \log(y^{-4}) \).Understanding this rule is key to further breaking down the expression into simpler parts.

Logarithmic Expansion

Logarithmic Expansion involves using properties of logarithms to break down complex logarithmic expressions into simpler, more manageable parts. After applying the Power Rule and Product Rule, you might encounter terms that are still not fully simplified. That's where logarithmic expansion comes in handy to simplify each part as much as possible.In our exercise, the steps included:

• Applying the Power Rule individually to terms: \( \log(x^3) = 3\log(x) \)and \( \log(y^{-4}) = -4\log(y) \).
• Distributing and simplifying the fraction: From \( \frac{1}{2}(3\log(x) - 4\log(y)) \)to \( \frac{3}{2}\log(x) - 2\log(y) \).

By the end of the process, the expression is fully expanded. Each step reduces complexity and gives a clearer view of the log's behavior in the expression. Through this methodology, handling logs in mathematical problems becomes much more manageable!