Problem 32 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to expand the expression. $$ \ln \frac{3 x^{2}}{(x+1)^{10}} $$

Step-by-Step Solution

Answer

\( \ln 3 + 2\ln x - 10\ln(x+1) \)

Step 1: Identify the Structure

The given expression is \( \ln \frac{3x^2}{(x+1)^{10}} \). It is a logarithm of a fraction, which suggests using the quotient rule in logarithms.

Step 2: Apply the Quotient Rule for Logarithms

According to the quotient rule, \( \ln \frac{a}{b} = \ln a - \ln b \). We can apply this to rewrite the expression as \( \ln 3x^2 - \ln (x+1)^{10} \).

Step 3: Expand the Logarithm of a Product

Next, we apply the product rule \( \ln (ab) = \ln a + \ln b \) to the term \( \ln 3x^2 \), which can be expanded to \( \ln 3 + \ln x^2 \).

Step 4: Apply the Power Rule for Logarithms

The power rule states that \( \ln a^n = n \ln a \). Apply this to \( \ln x^2 \), resulting in \( 2\ln x \), and to \( \ln (x+1)^{10} \), resulting in \( 10 \ln (x+1) \).

Step 5: Combine Results

Combine all the expanded parts to get the final expression. Therefore, the expanded form is \( \ln 3 + 2\ln x - 10\ln(x+1) \).

Key Concepts

Quotient Rule of Logarithms

The Quotient Rule of Logarithms is a fundamental concept that helps in the simplification and expansion of logarithmic expressions involving fractions. When you have a logarithm of a quotient, such as \( \ln \frac{a}{b} \), the rule offers a way to break it down into a subtraction of logarithms. It states that \( \ln \frac{a}{b} = \ln a - \ln b \). This is very useful because it allows us to separate and manage the terms more easily.

Consider the original expression \( \ln \frac{3x^2}{(x+1)^{10}} \). Here, we can see the fraction, which immediately signals the usage of the quotient rule. By applying the rule, we break it down into \( \ln 3x^2 - \ln (x+1)^{10} \). By transforming the log of a quotient into the difference of two logs, we are better positioned to apply further rules and simplify the expression even more.

This rule highlights how logarithms transform division inside a log into subtraction, making complex operations more straightforward.

Product Rule of Logarithms

The Product Rule of Logarithms is another essential tool in the toolkit for expanding logarithmic expressions. When you encounter the logarithm of a product, such as \( \ln (ab) \), this rule tells us it can be expressed as the sum of two separate logarithms: \( \ln a + \ln b \).

In our example, after applying the quotient rule, we encounter the term \( \ln 3x^2 \). This is a suitable situation for the product rule because it can be seen as the product of two separate components, 3 and \( x^2 \). By using the product rule, we expand \( \ln 3x^2 \) into \( \ln 3 + \ln x^2 \).

This ability to separate products within a logarithm aids in understanding and further manipulation of the expression. It is particularly helpful when dealing with more complex or composite numbers, allowing each part to be handled individually.

Power Rule of Logarithms

The Power Rule of Logarithms provides a straightforward method for dealing with logarithms of expressions raised to a power. It states that \( \ln a^n = n \ln a \). This rule simplifies expressions by moving the exponent to the front of the logarithmic expression as a multiplier.

In our problem, we apply the power rule to simplify \( \ln x^2 \) and \( \ln (x+1)^{10} \). For \( \ln x^2 \), using the power rule results in \( 2\ln x \). For \( \ln (x+1)^{10} \), it becomes \( 10\ln (x+1) \).

By transforming expressions with powers into a multiplication, the power rule makes it easier to manipulate and understand logarithmic terms. This aids in reducing the complexity of expressions, making it an indispensable rule in logarithmic operations.