Problem 39 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\( \frac{1}{4} \log(x^{2}+y^{2}) \)

Step 1: Understand the Given Expression

The given expression is \( \log \sqrt[4]{x^{2}+y^{2}} \). We need to expand this using logarithmic rules. The expression inside the logarithm is a fourth root, and it's important to recognize this to apply the correct logarithmic laws.

Step 2: Convert the Root to an Exponent

Using the property of roots, recall that \( \sqrt[4]{a} = a^{1/4} \). So, we can rewrite the expression inside the logarithm as \( (x^2 + y^2)^{1/4} \).

Step 3: Apply the Power Rule of Logarithms

The Power Rule of Logarithms states that \( \log(a^b) = b \cdot \log(a) \). Apply this rule to the expression \( \log((x^2 + y^2)^{1/4}) \). This becomes \( \frac{1}{4} \cdot \log(x^2 + y^2) \).

Step 4: Verify the Expansion

Verify each application of the logarithmic rules is correctly applied. In this example, no further expansion is available because \( x^2 + y^2 \) cannot be simplified further using logarithm laws in its current form.

Key Concepts

Power Rule of Logarithms

The Power Rule of Logarithms simplifies expressions involving terms raised to a power within a logarithmic function. This rule states: \( \log(a^b) = b \cdot \log(a) \). Essentially, you can "bring down" the exponent in front of the logarithm. This is very useful for simplifying complex expressions.Imagine you have an expression like \( \log((x^2 + y^2)^{1/4}) \). Here, the expression inside the logarithm is raised to the 1/4 power. The Power Rule allows you to take that exponent of 1/4 and multiply it by \( \log(x^2 + y^2) \), resulting in \( \frac{1}{4} \cdot \log(x^2 + y^2) \). This action simplifies the expression significantly, making further manipulations easier.In practice, the Power Rule helps when dealing with expressions where the base itself cannot be simplified with the logarithm laws, such as \( x^2 + y^2 \) in this particular example. It ensures you can still work with the expression efficiently.

Logarithmic Expansion

Logarithmic expansion is the process of breaking down complex logarithmic expressions into simpler parts using logarithmic laws. It's like unfolding a complicated function into more manageable pieces. This is particularly helpful for problems involving multiplication, division, or exponentiation inside a logarithm.When you are faced with an expression like \( \log((x^2 + y^2)^{1/4}) \), expansion can be applied after using the Power Rule. While \( x^2 + y^2 \) itself doesn't separate further with basic log rules, applying them to more involved expressions can ease computational work.Key laws used in logarithmic expansion include:

• The Product Rule: \( \log(a \times b) = \log(a) + \log(b) \)
• The Quotient Rule: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \)
• The Power Rule, as previously discussed

These rules simplify complex expressions into sums, differences, and multiples of simpler logarithms, making them easier to compute or integrate further in mathematical applications.

Roots and Exponents

Roots and exponents are fundamental concepts in mathematics, deeply interwoven with logarithms. Understanding their relationship is crucial when working with logarithmic functions.A root, for instance, can always be converted into an exponent. The expression \( \sqrt[4]{x^2 + y^2} \) is equivalently written as \( (x^2 + y^2)^{1/4} \). Here, the fourth root becomes an exponent of 1/4.This conversion is vital because logarithmic rules, especially the Power Rule, greatly rely on exponentials. Without transforming the root into an exponent, you cannot effectively apply the Power Rule to simplify expressions. Thus, grasping the method to transition between roots and exponentials allows for better manipulation of expressions involved with logarithms.Additionally, working with roots and exponents means recognizing the inverse relationship between these operations and their respective logarithmic functions, which can also aid in solving equations involving logarithms.