Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-10, simplify the given expression. \(\ln \left(x^{3} e^{-3 x}\right)\)

Step-by-Step Solution

Answer

The expression simplifies to \(3\ln(x) - 3x\).

Step 1: Apply the Logarithmic Identity

Recall that the logarithmic identity for the natural logarithm function states that \( \ln(a \cdot b) = \ln(a) + \ln(b) \). In the given expression \( \ln(x^{3} e^{-3x}) \), identify \( a = x^{3} \) and \( b = e^{-3x} \). Utilize this identity to separate the expression into two logarithms.

Step 2: Separate the Logarithms

Using the identity from Step 1, we can write \( \ln(x^{3} e^{-3x}) = \ln(x^{3}) + \ln(e^{-3x}) \). This separates the logarithm of the product into the sum of two logarithms.

Step 3: Simplify Each Logarithm

We'll simplify each part separately. Start with \( \ln(x^{3}) \). Through the power rule \( \ln(a^b) = b\ln(a) \), it simplifies to \( 3\ln(x) \). Next, for \( \ln(e^{-3x}) \), use the rule \( \ln(e^a) = a \) to get \( -3x \).

Step 4: Combine the Terms

After simplifying each logarithm, combine them into a single expression: \( 3\ln(x) - 3x \). This is the simplified form of the original expression.

Key Concepts

Natural Logarithm

The natural logarithm, often abbreviated as "ln," is a logarithm that has a special number, Euler's number \( e \), as its base. Euler's number is approximately equal to 2.71828. The natural logarithm is widely used in calculus and scientific calculations because of its unique properties and relationships to exponential functions.
Natural logarithms have a few important characteristics:

• \(\ln(e) = 1\), because \( e^1 = e \).
• \(\ln(1) = 0\), because \( e^0 = 1 \).
• The natural logarithm is undefined for non-positive numbers, as no power of \( e \) can result in a negative or zero value.

Understanding the natural logarithm is crucial when simplifying expressions involving exponential functions, as it allows us to transform and simplify using its inverses.

Logarithm Properties

Logarithms, including natural logarithms, have several essential properties that help in manipulating and simplifying expressions.
These properties are fundamental when dealing with complex mathematical expressions:

• Product Rule: \(\ln(a \cdot b) = \ln(a) + \ln(b)\). This lets us split the logarithm of a product into a sum of logarithms.
• Quotient Rule: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\). Here, the logarithm of a quotient is divided into a difference of two logs.
• Power Rule: \(\ln(a^b) = b \cdot \ln(a)\). This expresses the logarithm of an exponent as a product.

These properties are particularly useful for breaking down complex expressions into simpler forms. They help in reducing computational complexity and making calculations more manageable.

Simplification Techniques

Simplification techniques involve using the properties of logarithms to break down and simplify expressions. By doing this, we can transform complex expressions into more manageable forms.
In the case of the exercise, we utilize several steps:

• Use the Product Rule to break down the expression: \(\ln(x^3 e^{-3x}) = \ln(x^3) + \ln(e^{-3x})\).
• Apply the Power Rule on \(\ln(x^3)\): it simplifies to \(3\ln(x)\).
• Utilize the fact that \(\ln(e^a) = a\) to simplify \(\ln(e^{-3x})\) to \(-3x\).
• Combine all the simplified terms together to get the final expression: \(3\ln(x) - 3x\).

These techniques effectively reduce the complexity of the expression, making it easier to handle. Familiarity with such simplification methods empowers students to face a variety of problems more confidently.