Problem 43 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The expanded expression is \( 3 \ln(x) + \frac{1}{2} \ln(x-1) - \ln(3x+4) \).

Step 1: Apply the Quotient Rule

The Quotient Rule for logarithms states that \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \). Apply this to the expression \( \ln \left( \frac{x^3 \sqrt{x-1}}{3x+4} \right) \) to write it as \( \ln(x^3 \sqrt{x-1}) - \ln(3x+4) \).

Step 2: Apply the Product Rule

The Product Rule for logarithms states that \( \ln(ab) = \ln(a) + \ln(b) \). Apply this to the expression \( \ln(x^3 \sqrt{x-1}) \), which gives \( \ln(x^3) + \ln(\sqrt{x-1}) \).

Step 3: Simplify Using Power Rule

The Power Rule for logarithms states that \( \ln(a^b) = b \ln(a) \). Apply it to both components: \( \ln(x^3) = 3 \ln(x) \) and \( \ln(\sqrt{x-1}) = \ln((x-1)^{1/2}) = \frac{1}{2} \ln(x-1) \).

Step 4: Combine All Simplified Parts

Combine the expressions from previous steps:\( 3 \ln(x) + \frac{1}{2} \ln(x-1) - \ln(3x+4) \) is the expanded form of the original logarithmic expression.

Key Concepts

Quotient Rule

The Quotient Rule is an essential tool when dealing with logarithms, especially when you encounter a division within the logarithm's argument. The rule states: \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \). This property reflects how division inside a logarithm translates into subtraction between two logs. For instance, when we see an expression like \( \ln \left( \frac{x^3 \sqrt{x-1}}{3x+4} \right) \), applying the Quotient Rule allows us to break it down into \( \ln(x^3 \sqrt{x-1}) - \ln(3x+4) \). By doing this, we simplify the problem into smaller pieces that are easier to handle.

Using this rule effectively turns a complex division problem into a separate subtraction problem, helping to focus on each part individually. This transformation is central to breaking down and solving logarithmic equations.

Product Rule

The Product Rule is another powerful logarithmic tool that simplifies expressions involving multiplication. According to the Product Rule, \( \ln(ab) = \ln(a) + \ln(b) \). When you're working with logarithms, multiplication inside the logarithm becomes addition outside, making separate components more manageable. Consider \( \ln(x^3 \sqrt{x-1}) \) in our example. Applying the Product Rule, we can express this as \( \ln(x^3) + \ln(\sqrt{x-1}) \).

This step is particularly helpful because it further breaks down the problem, allowing us to apply other rules more easily. By translating the multiplication inside the log into addition outside, we create pathways to apply further rules, like the Power Rule, making the entire expression simpler step by step.

Power Rule

The Power Rule of logarithms helps manage expressions where a power exists within the logarithm. This rule is stated as \( \ln(a^b) = b \ln(a) \), and it's invaluable whenever exponents are involved. In breaking down something like \( \ln(x^3) \), you apply the Power Rule to move the exponent to the front, converting it to \( 3 \ln(x) \). Similarly, an expression such as \( \ln(\sqrt{x-1}) \) is treated as \( \ln((x-1)^{1/2}) \), and applying the Power Rule gives \( \frac{1}{2} \ln(x-1) \).

By applying the Power Rule, cumbersome exponent components within logs become straightforward multipliers in front, simplifying the entire notation. This transformation is crucial when our goal is to expand and simplify complex logarithmic expressions, making the end result much clearer and easier to evaluate.