Problem 41 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to expand the expression. $$\log \sqrt{\frac{x^{2}+4}{\left(x^{2}+1\right)\left(x^{3}-7\right)^{2}}}$$

Step-by-Step Solution

Answer

\( \frac{1}{2} \cdot \log(x^{2}+4) - \frac{1}{2} \cdot \log(x^{2}+1) - \log(x^{3}-7) \)

Step 1: Convert Root to Exponent

The expression inside the logarithm is a square root. By using the property \( \sqrt{a} = a^{1/2} \), we can rewrite the expression as: \[\log \left(\frac{x^{2}+4}{(x^{2}+1)(x^{3}-7)^{2}}\right)^{1/2}\] We bring the exponent out, using the property \( \log(a^b) = b \cdot \log(a) \):\[\frac{1}{2} \cdot \log \left(\frac{x^{2}+4}{(x^{2}+1)(x^{3}-7)^{2}}\right)\]

Step 2: Apply the Quotient Rule

The logarithm of a quotient gets expanded to the difference of logarithms, i.e., \( \log \left( \frac{a}{b} \right) = \log(a) - \log(b) \). Applying this rule, we get:\[\frac{1}{2} \cdot \left( \log(x^{2}+4) - \log((x^{2}+1)(x^{3}-7)^{2}) \right)\]

Step 3: Apply the Product Rule

The logarithm of a product is the sum of the logarithms, i.e., \( \log(a \cdot b) = \log(a) + \log(b) \). Apply this to the denominator:\[\frac{1}{2} \cdot \left( \log(x^{2}+4) - ( \log(x^{2}+1) + \log((x^{3}-7)^2) ) \right)\]

Step 4: Simplify Using the Power Rule

Using the power rule for logarithms again on \( \log((x^3-7)^2) \), we get \( 2 \cdot \log(x^3-7) \). Substitute this back into the expression:\[\frac{1}{2} \cdot \left( \log(x^{2}+4) - \log(x^{2}+1) - 2 \cdot \log(x^{3}-7) \right)\]

Step 5: Distribute the Fraction

Finally, distribute \( \frac{1}{2} \) into the expression to complete the expansion:\[\frac{1}{2} \cdot \log(x^{2}+4) - \frac{1}{2} \cdot \log(x^{2}+1) - \log(x^{3}-7)\]

Key Concepts

Logarithmic Expressions

Understanding logarithmic expressions is fundamental in mastering logarithms. A logarithmic expression typically involves finding the power to which a number (the base) must be raised to get another number. The expression given in the problem is:\[\log \sqrt{\frac{x^{2}+4}{\left(x^{2}+1\right)\left(x^{3}-7\right)^{2}}}\]This expression involves natural and mathematical components such as roots, fractions, and powers, represented within a logarithm. Understanding the role of these components helps in applying the laws of logarithms effectively. In this context, the square root means that the expression inside the root is raised to the power of 1/2. By successfully manipulating these components using the laws of logarithms, we transform complex mathematical statements into simpler, more manageable forms.

Expansion of Logarithms

Expanding a logarithmic expression involves using the properties of logarithms to break down a complex expression into simpler parts. For the expression given, we utilized several rules to achieve expansion:

• Root to Exponent Conversion: Recognize that the square root translates to a power of 1/2.
• Quotient Rule: Separate the division inside the log as a subtraction between two logarithms.
• Product Rule: Break down products in the denominator into a sum of logs.
• Power Rule: Remove powers by multiplying the logarithm by the exponent.

By applying these rules systematically, the expression becomes:\[\frac{1}{2} \cdot \log(x^{2}+4) - \frac{1}{2} \cdot \log(x^{2}+1) - \log(x^{3}-7)\]Expansion allows for detailed analysis and makes complex expressions more approachable by reducing them to basic log identities.

Logarithmic Identities

Logarithmic identities are the essential tools for operating on and simplifying logarithmic expressions. In our problem, we relied on several key identities:

• Quotient Rule: This states that \( \log(a/b) = \log(a) - \log(b) \), allowing us to separate the numerator and denominator.
• Product Rule: This identity explains that \( \log(ab) = \log(a) + \log(b) \), crucial when dealing with products in our expressions.
• Power Rule: This simplifies \( \log(a^b) \) into \( b \cdot \log(a) \), used to break down powers.

These identities convert complex formulas into understandable expressions by breaking them into parts. Mastery of these rules is indispensable in both expanding and contracting logarithmic expressions, providing a foundation for advanced problem-solving in mathematics.