Problem 34 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to expand the expression. $$\log \left(\frac{a^{2}}{b^{4} \sqrt{c}}\right)$$

Step-by-Step Solution

Answer

\(2\log(a) - 4\log(b) - \frac{1}{2}\log(c)\)

Step 1: Apply the Quotient Rule

According to the quotient law of logarithms, \( \log \left( \frac{x}{y} \right) = \log(x) - \log(y) \). Here, apply this to split the log into subtraction: \( \log \left(\frac{a^{2}}{b^{4} \sqrt{c}}\right) = \log(a^{2}) - \log(b^{4} \sqrt{c}) \).

Step 2: Expand the Product in the Denominator

Use the product law of logarithms, \( \log(xy) = \log(x) + \log(y) \), to expand \( \log(b^{4} \sqrt{c}) \). This gives: \( \log(b^{4} \sqrt{c}) = \log(b^{4}) + \log(\sqrt{c}) \).

Step 3: Apply the Power Rule to Each Logarithm

Utilize the power rule, \( \log(x^n) = n\log(x) \). First, expand \( \log(a^{2}) = 2\log(a) \). For the denominator terms, expand \( \log(b^{4}) = 4\log(b) \) and \( \log(\sqrt{c}) = \log(c^{1/2}) = \frac{1}{2}\log(c) \).

Step 4: Substitute Back and Finalize

Substitute all the expanded terms back: \( \log \left(\frac{a^{2}}{b^{4} \sqrt{c}}\right) = 2\log(a) - \left(4\log(b) + \frac{1}{2}\log(c)\right) = 2\log(a) - 4\log(b) - \frac{1}{2}\log(c) \).

Key Concepts

Quotient Rule

The Quotient Rule in logarithms is a powerful tool that helps us when dealing with the division of numbers inside a logarithm. When faced with an expression like \( \log \left( \frac{x}{y} \right) \), the Quotient Rule allows us to rewrite this as \( \log(x) - \log(y) \). This means we can separate the logarithm into a difference between two simpler logarithms. It simplifies complex logarithmic expressions and is especially useful in algebraic manipulations.

Think of the Quotient Rule as a way to "unpack" division inside a log into two separate, easier-to-handle pieces. This reduction from division to subtraction is critical for simplifying and expanding logarithmic expressions. It makes it easier to manage and solve problems involving complex equations involving logs.

Product Rule

The Product Rule of logarithms is used to break down multiplicative relationships within a logarithmic expression. According to this rule, \( \log(xy) = \log(x) + \log(y) \). This means if you have a product inside a logarithm, you can "split" it into the sum of two individual logarithms.

This rule is particularly helpful when simplifying or expanding expressions, as it converts multiplication into addition. By applying the Product Rule, we can easily manage complex products within logs. For instance, in the expression \( \log(b^{4} \sqrt{c}) \), the product \( b^{4} \sqrt{c} \) can be expanded into \( \log(b^{4}) + \log(\sqrt{c}) \). It makes otherwise cumbersome calculations clear and straightforward.

Power Rule

When you encounter a power inside a logarithmic expression, such as \( \log(x^n) \), the Power Rule allows you to extract the exponent and multiply it with the logarithm as \( n\log(x) \). This rule simplifies expressions by moving the exponent in front, reducing the complexity of the calculation.

For example, using the Power Rule, we can deal with expressions like \( \log(a^2) \) and \( \log(b^4) \) by rewriting them as \( 2\log(a) \) and \( 4\log(b) \), respectively. Even roots can be rewritten using fractional exponents, so \( \log(\sqrt{c}) = \log(c^{1/2}) = \frac{1}{2}\log(c) \). This rule transforms potentially unwieldy expressions into more manageable forms, enabling easier computation and understanding.