Problem 27 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{3}(x \sqrt{y}) $$

Step-by-Step Solution

Answer

\( \log_{3}(x) + \frac{1}{2} \log_{3}(y) \)

Step 1: Understand the Expression

The given expression is \( \log_{3}(x \sqrt{y}) \). We need to use the Laws of Logarithms to expand this expression. The key laws we will use are: the Product Rule and the Power Rule of logarithms.

Step 2: Apply the Product Rule

The Product Rule states that \( \log_b(MN) = \log_b(M) + \log_b(N) \). We can consider \( x \sqrt{y} \) as a product of \( x \) and \( \sqrt{y} \), and apply the Product Rule: \[ \log_{3}(x \sqrt{y}) = \log_{3}(x) + \log_{3}(\sqrt{y}) \]

Step 3: Simplify the Radical Using the Power Rule

The Power Rule states that \( \log_b(M^n) = n \cdot \log_b(M) \). Recognize that \( \sqrt{y} = y^{1/2} \). Use the Power Rule to expand \( \log_{3}(\sqrt{y}) \): \[ \log_{3}(\sqrt{y}) = \log_{3}(y^{1/2}) = \frac{1}{2} \cdot \log_{3}(y) \]

Step 4: Combine the Results

Substitute the expanded form from Step 3 back into the expression we obtained in Step 2: \[ \log_{3}(x) + \frac{1}{2} \log_{3}(y) \]. This gives us the expanded form of the original logarithmic expression.

Key Concepts

Product Rule of Logarithms

The Product Rule of Logarithms is a fundamental rule that helps in expanding logarithmic expressions involving a product. When you come across a logarithm of a product, the Product Rule indicates that you can split it into the sum of separate logarithms. This is based on the identity:

• \( \log_b(MN) = \log_b(M) + \log_b(N) \)

In simpler terms, if you're working with \( \log_b(x \cdot y) \), you can express this as two separate parts: \( \log_b(x) \) plus \( \log_b(y) \).

Consider our exercise expression \( \log_{3}(x \sqrt{y}) \). Here, \( x \sqrt{y} \) is a product, so we apply the Product Rule:

• First, identify the two parts: \( x \) and \( \sqrt{y} \).
• Then, expand it using the Product Rule: \( \log_{3}(x) + \log_{3}(\sqrt{y}) \).

This expansion is the first step in breaking down complex logarithmic expressions into simpler components that are easier to handle in further calculations.

Power Rule of Logarithms

The Power Rule is another important tool when working with logarithms, especially useful when dealing with expressions raised to a power. The Power Rule states:

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

This means if you have a power inside a logarithm, you can "bring it down" to the front as a multiplying factor of the logarithm.

In our exercise, we use the Power Rule for simplifying the radical \( \sqrt{y} \). Notice that \( \sqrt{y} \) is the same as \( y^{1/2} \). Thus, applying the Power Rule:

• \( \log_{3}(y^{1/2}) = \frac{1}{2} \cdot \log_{3}(y) \)

This step transforms the expression by turning the exponent into a fraction that multiplies the log. It provides a way to handle roots or powers within logarithmic expressions by converting them into manageable forms.

Logarithmic Expression Expansion

Logarithmic Expression Expansion involves breaking down complex logarithmic expressions into simpler components, often using laws like the Product and Power Rules. By expanding expressions, you can make the calculations more manageable and simplify understanding.

In the given exercise, we start with \( \log_{3}(x \sqrt{y}) \), aiming to expand it fully:

• First, apply the Product Rule, dividing the expression into \( \log_{3}(x) + \log_{3}(\sqrt{y}) \).
• Next, deal with the \( \sqrt{y} \) using the Power Rule, which gives us \( \frac{1}{2} \log_{3}(y) \).
• This results in a completely expanded form: \( \log_{3}(x) + \frac{1}{2} \log_{3}(y) \).

Expanding logarithmic expressions is a crucial skill, particularly in algebra and calculus, as it simplifies solving equations and understanding relationships between different components of the expression.