Problem 28 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the laws of logarithms to expand and simplify the expression. $$\ln \frac{x^{2}}{\sqrt{x}(1+x)^{2}}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the logarithmic laws to apply

In this step, we'll identify the logarithmic laws to apply on the given logarithmic expression. The expression can be rewritten using a combination of the logarithm properties: 1. The quotient rule (log_a(b/c) = log_a b - log_a c) 2. The power rule (log_a b^c = c*log_a b) 3. The product rule (log_a(bc) = log_a b + log_a c) The given expression is: \(\ln \frac{x^{2}}{\sqrt{x}(1+x)^{2}}\)

Step 2: Apply the quotient rule

Apply the quotient rule on the given expression. The quotient rule states that log_a(b/c) = log_a b - log_a c. In our case, a = e, b = x^2, and c = \(\sqrt{x}(1+x)^{2}\). So, \(\ln \frac{x^{2}}{\sqrt{x}(1+x)^{2}} = \ln(x^{2}) - \ln(\sqrt{x}(1+x)^{2})\)

Step 3: Apply the power and product rules

Now, apply the power rule and product rule to further expand the expression. First, apply the power rule to \(\ln(x^{2})\): \(2 \times \ln{x}\) Second, apply the product rule to \(\ln(\sqrt{x}(1+x)^{2})\): \(\ln\sqrt{x} + \ln(1+x)^{2}\) Now, our expression looks like: \(2\ln{x} - (\ln\sqrt{x} + \ln(1+x)^{2})\)

Step 4: Apply the power rule again

Apply the power rule again on \(\ln\sqrt{x}\) and \(\ln(1+x)^{2}\). For \(\ln\sqrt{x}\), the power rule gives us: \(\frac{1}{2} \times \ln{x}\) For \(\ln(1+x)^{2}\), the power rule gives us: \(2 \times \ln(1+x)\) Our expression now looks like: \(2\ln{x} - (\frac{1}{2}\ln{x} + 2\ln(1+x))\)

Step 5: Simplify the expression

Now, simplify the expression by combining the terms: \(2\ln{x} - \frac{1}{2}\ln{x} - 2\ln(1+x)\) Combine the first two terms: \((2 - \frac{1}{2})\ln{x} - 2\ln(1+x)\) \(= \frac{3}{2}\ln{x} - 2\ln(1+x)\) The expanded and simplified expression is: \(\frac{3}{2}\ln{x} - 2\ln(1+x)\)

Key Concepts

Quotient Rule

The quotient rule for logarithms is a handy tool when dealing with the logarithm of a division of numbers. It states that:
\[ \log_a \left( \frac{b}{c} \right) = \log_a b - \log_a c \]
This means you can separate the natural log of a quotient into the difference of two logs. It's like saying if you have a pie, you can think of it in terms of pie slices where each piece stands for a part of \(b\) or \(c\). In the context of our exercise, the expression \(\ln \frac{x^{2}}{\sqrt{x}(1+x)^{2}}\) is simplified into two natural logs by splitting the fraction:

• First is the log of the numerator, \(\ln(x^2)\).
• Second is the subtraction of the log of the denominator, \(\ln(\sqrt{x}(1+x)^2)\).

By applying the quotient rule, we made the expression simpler and easier for further expansion using other log rules.

Power Rule

The power rule of logarithms allows us to work with exponents inside a logarithm by moving them outside as a multiplier. This rule is expressed as:
\[ \log_a b^c = c \cdot \log_a b \]
For example, \(\ln(x^2)\) applies the power rule to shift the \(2\) in the exponent to become \(2 \cdot \ln x\). Similarly, when we encounter \(\ln(1+x)^2\), the exponent \(2\) is moved outside as \(2 \cdot \ln(1+x)\). Additionally, for expressions with roots like \(\ln(\sqrt{x})\), we interpret this as a power of \(\frac{1}{2}\), which converts to \(\frac{1}{2} \cdot \ln x\) after applying the power rule.
This rule helps to manage terms with compound exponents, transforming them into simpler linear forms which ease the entire logarithmic process of simplification.

Product Rule

The product rule for logarithms is essential when dealing with the log of a multiplication. The rule is expressed as:
\[ \log_a(bc) = \log_a b + \log_a c \]
In our previous steps when simplifying \( \ln(\sqrt{x}(1+x)^2)\), this rule splits the product inside the log into a sum of logs:

• The first product, \( \ln(\sqrt{x}) \), becomes \( \frac{1}{2} \ln x \) after applying the power rule.
• Second, \( \ln(1+x)^2 \), after a power rule turns into \( 2 \ln(1+x) \).

The product rule is crucial because it breaks down multiplied expressions into manageable parts that are simpler to manipulate, making the overall solution clearer and more concise. This approach paves the way for a straightforward pathway to solve and simplify log equations.