Problem 43 - Algebra and Trigonometry

Question

\(19-44\) 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 logarithm of a quotient can be expressed as the difference of the logarithms: \[ \ln\left(\frac{A}{B}\right) = \ln(A) - \ln(B) \].Apply this to the expression: \[ \ln\left(\frac{x^{3} \sqrt{x-1}}{3x+4}\right) = \ln(x^{3} \sqrt{x-1}) - \ln(3x+4) \].

Step 2: Apply the Product Rule

The logarithm of a product can be expressed as the sum of the logarithms: \[ \ln(A \cdot B) = \ln(A) + \ln(B) \].Apply this to the term \( \ln(x^{3} \sqrt{x-1}) \):\[ \ln(x^{3} \sqrt{x-1}) = \ln(x^{3}) + \ln(\sqrt{x-1}) \].

Step 3: Simplify Logarithms of Powers and Roots

Use the power rule of logarithms which shows that if \( c \) is a constant, then \( \ln(a^c) = c \ln(a) \).For \( \ln(x^3) \), simplify it to:\[ \ln(x^3) = 3 \ln(x) \].For \( \ln(\sqrt{x-1}) \), recall that \( \sqrt{x-1} = (x-1)^{1/2} \) and simplify to:\[ \ln(\sqrt{x-1}) = \frac{1}{2}\ln(x-1) \].

Step 4: Combine all Components

Substitute the simplified components back into the expression from Step 1:\[ \ln\left(\frac{x^{3} \sqrt{x-1}}{3x+4}\right) = 3 \ln(x) + \frac{1}{2} \ln(x-1) - \ln(3x+4) \].This is the expanded form of the given logarithmic expression.

Key Concepts

Quotient Rule

The quotient rule is a handy tool when working with logarithmic expressions involving division. In simple terms, it tells us how to break down the logarithm of a fraction. Imagine you have a fraction, which is basically one number divided by another. The quotient rule states that you can split this into two separate logarithms: the log of the top number minus the log of the bottom number.
This rule is expressed mathematically as:

• \(\ln\left(\frac{A}{B}\right) = \ln(A) - \ln(B)\)

This rule is essential when you aim to expand logarithmic expressions. In our original exercise, this was applied to the expression \( \ln\left(\frac{x^{3}\sqrt{x-1}}{3x+4}\right) \). The entire equation was divided into two separate parts: \( \ln(x^3\sqrt{x-1}) \) and \( \ln(3x+4) \).
Remember, the quotient rule is all about subtraction when it comes to splitting the logarithm of a fraction. This makes it incredibly useful for simplifying complex logarithmic expressions by breaking them down into more manageable parts.

Product Rule

The product rule comes into play when we have a logarithm of a product, meaning two numbers or expressions multiplied together. This rule helps us to expand logarithmic expressions involving multiplication. According to the product rule, the logarithm of a product is the sum of the logarithms of each factor.
The product rule can be written as:

• \(\ln(A \cdot B) = \ln(A) + \ln(B)\)

In the exercise, this was applied to the term \( \ln(x^3 \sqrt{x-1}) \), breaking it down into \( \ln(x^3) + \ln(\sqrt{x-1}) \).
The product rule helps transform a single, intricate logarithmic expression into smaller and simpler parts, by making each multiplicative component its own separate term in the expansion.

Power Rule

The power rule is all about dealing with exponents inside logarithms. This rule simplifies how we handle scenarios when there's a power or a root within a logarithmic expression. The rule states that any exponent in a log expression can be brought out as a coefficient in front of the logarithm.
Mathematically, the power rule is expressed as:

• \(\ln(a^c) = c \ln(a)\)

In the case of our exercise, this rule was crucial. For the term \( \ln(x^3) \), using the power rule, it becomes \( 3 \ln(x) \). Similarly, for \( \ln(\sqrt{x-1}) \), which is \( \ln((x-1)^{1/2}) \), it is rewritten as \( \frac{1}{2} \ln(x-1) \) by recognizing the square root as a power of \( 1/2 \).
This rule makes it straightforward to manage exponents in logarithmic expressions, turning complex powers into simple multipliers that are much easier to handle in calculations.