Problem 33 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to expand the expression. $$ \log \sqrt[4]{x^{2}+y^{2}} $$

Step-by-Step Solution

Answer

\( \frac{1}{4} \log(x^2 + y^2) \) is the expanded form.

Step 1: Apply the Power Rule of Logarithms

Start by expressing the expression under the logarithm as a power. Since the expression is a fourth root, rewrite \( \sqrt[4]{x^2 + y^2} \) as \((x^2 + y^2)^{1/4}\). The power rule of logarithms states \( \log(a^b) = b \log(a) \). Using this, we can expand \( \log((x^2 + y^2)^{1/4}) \) to \( \frac{1}{4} \log(x^2 + y^2) \).

Step 2: Apply the Sum Rule of Logarithms

Now expand \( \log(x^2 + y^2) \). However, the expression \( x^2 + y^2 \) inside the logarithm cannot be split further using the rules of logarithms since logarithms only split multiplication into a sum of terms, not addition into separate terms. We have to retain it as is.

Key Concepts

Power Rule of Logarithms

The Power Rule of Logarithms is an essential mathematical tool that allows us to simplify logarithmic expressions involving exponents. When you have a logarithm of an exponential expression, such as \( \log(a^b) \), the Power Rule states that you can bring the exponent \( b \) in front of the logarithm as a multiplier. This transforms the expression to \( b \log(a) \). To illustrate, let's look at an example: suppose we need to simplify \( \log((x^2 + y^2)^{1/4}) \). First, recognize that the fourth root can be rewritten as an exponent, specifically \((x^2 + y^2)^{1/4}\). By applying the Power Rule, this expression simplifies to \( \frac{1}{4} \log(x^2 + y^2) \). Using the Power Rule makes complex expressions more manageable and is especially useful when dealing with roots, which can always be expressed as fractional exponents. This reduction is important because it prepares the expression for potential further simplification or evaluation, depending on the mathematical context.

Sum Rule of Logarithms

The Sum Rule of Logarithms, often called the Addition Rule, allows us to expand a logarithm of a product into the sum of logarithms of its factors. The rule states that \( \log(a \times b) = \log(a) + \log(b) \). This rule is helpful in breaking down products inside a logarithm into simpler, more digestible parts.In the given exercise, we encounter the term \( \log(x^2 + y^2) \) after applying the Power Rule. However, it's crucial to understand the limitations of the Sum Rule of Logarithms here. The expression \( x^2 + y^2 \) is a sum, not a product, and therefore cannot be separated into individual logarithmic terms using the Sum Rule. The rule exclusively applies to multiplication inside the logarithm, not addition. Appreciating why certain terms cannot be further expanded under these rules prevents misconceptions and helps in correctly simplifying logarithmic expressions without unnecessary errors.

Logarithmic Expansion

Logarithmic Expansion refers to the process of using the Laws of Logarithms to break down complex logarithmic expressions into simpler parts. This can involve the application of the Power, Sum, and Difference Rules, based on the structure of the given expression.In the context of our exercise, the initial expression \( \log \sqrt[4]{x^2+y^2} \) is expanded using the Power Rule, resulting in \( \frac{1}{4} \log(x^2 + y^2) \). This step-by-step transformation illustrates the concept of logarithmic expansion—turning a complicated logarithmic form into one that is easier to evaluate or further manipulate.However, the expansion is bounded by the inherent restrictions of logarithmic properties. Since the components inside the logarithm in the exercise are additions, not products, we refrain from using the Sum Rule to further expand. Logarithmic Expansion is a powerful mathematical technique, but it's essential to apply the correct rules and recognize when an expression has reached its fully expanded form. This skill in recognizing the limits and possibilities of expansion is crucial for simplifying and solving logarithmic expressions efficiently.