Problem 37 - Algebra and Trigonometry

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 Logarithm of a Quotient Rule

The first step is to separate the numerator and the denominator using the quotient rule of logarithms, which states that \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \). Thus, we write: \[\ln\left(\frac{x^3 \sqrt{x-1}}{3x+4}\right) = \ln\left(x^3 \sqrt{x-1}\right) - \ln(3x+4) \]This results in two separate logarithmic expressions.

Step 2: Apply the Logarithm of a Product Rule

The next step is to expand the logarithm of the numerator, \( \ln\left(x^3 \sqrt{x-1}\right) \), using the product rule of logarithms, \( \ln(ab) = \ln(a) + \ln(b) \). In this case, let \( a = x^3 \) and \( b = \sqrt{x-1} \):\[\ln\left(x^3 \sqrt{x-1}\right) = \ln(x^3) + \ln(\sqrt{x-1})\]

Step 3: Apply the Power Rule of Logarithms

Use the power rule of logarithms which states that \( \ln(a^b) = b\ln(a) \). First, apply it to \( \ln(x^3) \):\[\ln(x^3) = 3\ln(x)\]Next, apply the rule to \( \ln(\sqrt{x-1}) \), remembering \( \sqrt{x-1} = (x-1)^{1/2} \):\[\ln((x-1)^{1/2}) = \frac{1}{2}\ln(x-1)\]

Step 4: Combine All Parts

Now, combine all the parts from the previous steps to form the expanded expression:\[\ln\left(x^3 \sqrt{x-1}\right) - \ln(3x+4) = 3\ln(x) + \frac{1}{2}\ln(x-1) - \ln(3x+4)\]This is the expanded form of the logarithmic expression.

Key Concepts

Quotient Rule

Logarithms are incredibly useful for transforming complex operations into simpler ones. One vital logarithm rule is the Quotient Rule. This rule helps us separate a single fraction under a logarithm into two more manageable parts.
In basic terms, the Quotient Rule states: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \). This means you take the logarithm of the numerator and subtract the logarithm of the denominator.

• This subtraction is key—it simplifies multiplication and division problems into addition and subtraction.
• Using the rule, you help solve complex expressions by making them easier to handle.

For example, given an expression \( \ln\left(\frac{x^3 \sqrt{x-1}}{3x+4}\right) \), we first split it into two distinct logarithmic parts: \( \ln(x^3 \sqrt{x-1}) \) and \( \ln(3x+4) \). Each part now stands alone and can be further broken down using other rules.

Product Rule

Now that we've split our initial logarithmic expression using the Quotient Rule, our task is to deal with the top part \( \ln(x^3 \sqrt{x-1}) \). We employ the Product Rule of logarithms here, which is another vital tool.
The Product Rule is straightforward: for two multiplicants \( a \) and \( b \), you apply \( \ln(ab) = \ln(a) + \ln(b) \).

• It turns multiplication inside a logarithm into addition outside—another way of simplifying the expression.
• This rule is extremely helpful when dealing with compounded variables or expressions within a logarithm.

In our example, set:\( a = x^3 \) and \( b = \sqrt{x-1} \). Then, separate them using the product rule: \( \ln(x^3 \sqrt{x-1}) = \ln(x^3) + \ln(\sqrt{x-1}) \). This step was critical to make the expression more digestible for subsequent rules.

Power Rule

Having two separate logarithmic expressions lets us use the Power Rule. This particular rule offers direct handling of exponential terms inside a logarithm.
According to the Power Rule, if you have \( \ln(a^b) \), it becomes \( b \cdot \ln(a) \). You effectively bring down the exponent as a multiplier of the logarithm.

• This transformation makes it easier to deal with the variable.
• It simplifies expressions by reducing the complexity of powers.

In our example, the Power Rule breaks \( \ln(x^3) \) to \( 3\ln(x) \). Similarly, considering \( \sqrt{x-1} = (x-1)^{1/2} \), then \( \ln((x-1)^{1/2}) = \frac{1}{2}\ln(x-1) \). This adjustment greatly aids in fully expanding and understanding the logarithmic expression.

Expanding Logarithms

The real goal of applying these logarithm rules is to expand a complex expression into a simpler and more understandable form. Expanding logarithms breaks a single complicated logarithmic expression into several manageable pieces.
Here's why expanding is important:

• It helps solve real-world problems where inputs are not simple variables.
• Being able to expand logs can guide complex derivations and calculations.

With all the rules applied in our example, the full expanded form of the logarithmic expression \( \ln\left(\frac{x^3 \sqrt{x-1}}{3x+4}\right) \) becomes \( 3\ln(x) + \frac{1}{2}\ln(x-1) - \ln(3x+4) \).
This breakdown shows each component of the original expression and makes it clear how each part influences the final value. Understanding and applying these logarithm rules not only streamlines solving equations but also enriches general mathematical proficiency.