Problem 39 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \sqrt[4]{x^{2}+y^{2}} $$

Step-by-Step Solution

Answer

Expand as \( \frac{1}{4} \cdot \log(x^2 + y^2) \).

Step 1: Rewrite the Radical as an Exponent

First, we need to express the fourth root as an exponent. The expression \( \sqrt[4]{x^2 + y^2} \) can be rewritten using exponents as \( (x^2 + y^2)^{1/4} \). This allows us to use the properties of logarithms with exponents.

Step 2: Apply the Power Rule of Logarithms

The power rule for logarithms states that \( \log(a^b) = b \cdot \log(a) \). Using this rule, we can move the exponent \( \frac{1}{4} \) in front of the logarithm: \( \log((x^2 + y^2)^{1/4}) = \frac{1}{4} \cdot \log(x^2 + y^2) \).

Step 3: Final Expression

Now that the expression is expanded using the power rule, we have \( \frac{1}{4} \cdot \log(x^2 + y^2) \). This is the expanded form of the original logarithmic expression.

Key Concepts

Laws of Logarithms

Logarithms have specific rules that make them easier to work with. These are called the laws of logarithms and they help simplify complex logarithmic expressions. There are three main laws:

• Product Law: This law states that the logarithm of a product is equal to the sum of the logarithms of the factors. In formula form, \( \log_b(a \cdot c) = \log_b a + \log_b c \).
• Quotient Law: This law states that the logarithm of a quotient is equal to the difference of the logarithms. That is, \( \log_b\left( \frac{a}{c} \right) = \log_b a - \log_b c \).
• Power Rule: This law allows us to bring the exponent down in front of the logarithm, making calculations easier. It is expressed as \( \log_b(a^c) = c \cdot \log_b a \).

Understanding these laws is crucial for expanding logarithmic expressions effectively. They allow us to manipulate logarithmic expressions in a way that simplifies solving equations, integrating, or differentiating them. Each law finds wide application depending on the form of the expression you're dealing with.

Power Rule of Logarithms

The power rule of logarithms is particularly useful when dealing with logarithmic expressions that involve exponents. Essentially, it states that if you have a logarithm of a number raised to an exponent, you can simplify the expression by bringing the exponent to the front. This is expressed in the formula: \( \log_b(a^c) = c \cdot \log_b a \).
Applying this rule reduces the complexity of calculations by eliminating the exponent within the logarithm. Consider the expression \( \log((x^2 + y^2)^{1/4}) \). Using the power rule, you move the \( \frac{1}{4} \) in front of the logarithm, which gives you \( \frac{1}{4} \cdot \log(x^2 + y^2) \). This transformation simplifies the work needed to expand or further manipulate the expression.
The power rule is invaluable in both mathematical and real-world calculations, where it is necessary to manage large numerical exponents efficiently. Whether you're dealing with algebraic expressions or scientific computations, the power rule provides a straightforward means to streamline and simplify expressions.

Exponential Expressions

Exponential expressions, like the one seen in the problem \( (x^2 + y^2)^{1/4} \), involve variables raised to a power, which indicates repeated multiplication. Exponents are a way to represent numbers that multiply themselves a certain number of times. In this case, the expression involves taking the fourth root, equivalent to raising the expression to the power of \( \frac{1}{4} \).

• An exponential expression like \( x^n \) means that \( x \) is multiplied by itself \( n \) times.
• Expressions such as these are crucial building blocks in mathematics, appearing in algebra, calculus, and beyond.
• Understanding how to manipulate them using logarithmic rules allows for deeper insights into mathematical relationships and more robust equation-solving techniques.

When you rewrite roots as powers, you can effectively apply logarithmic rules, such as the power rule, to transform and simplify expressions. This ability to switch between different forms highlights the versatility of mathematical operations and their interconnected nature.