Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify the given expression. $$ \ln \left(x^{3} e^{-3 x}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( 3 \ln x - 3x \).

Step 1: Apply Logarithm Property

Use the property of logarithms: \( \ln(a \cdot b) = \ln a + \ln b \). This allows us to split the expression: \( \ln(x^3 e^{-3x}) = \ln(x^3) + \ln(e^{-3x}) \).

Step 2: Simplify Each Logarithm

Now, apply the property \( \ln(a^b) = b \ln a \). This simplifies the terms to: \( \ln(x^3) = 3 \ln x \) and \( \ln(e^{-3x}) = -3x \ln e \).

Step 3: Evaluate \(\ln e\)

Recall that \( \ln e = 1 \). Substitute \( 1 \) into the expression: \( -3x \ln e = -3x \times 1 = -3x \).

Step 4: Combine Results

Combine the simplified terms: \( 3 \ln x + (-3x) = 3 \ln x - 3x \).

Key Concepts

Logarithm Properties

When dealing with logarithmic expressions, understanding the properties of logarithms can greatly simplify complex calculations. One of the fundamental properties is the **Product Property** of logarithms. This states that for any positive numbers \(a\) and \(b\), the natural logarithm can be split as follows:

• \( \ln(ab) = \ln a + \ln b \)

Applying this property helps break down expressions into more manageable pieces. In the example given, \( \ln(x^3 e^{-3x}) = \ln(x^3) + \ln(e^{-3x}) \), demonstrates using this property effectively.
Another important property is the **Power Property**, which expresses that for any real number \(b\) and positive number \(a\), you can rewrite the logarithm as:

• \( \ln(a^b) = b \cdot \ln a \)

This property allows us to bring exponents down and simplifies into a linear multiplication problem. For example, in our case, this converts \(\ln(x^3)\) into \(3\ln x\) and \(\ln(e^{-3x})\) into \(-3x \ln e\).
Understanding these logarithm properties allows for smooth transitions from complex to simplified forms, making them indispensable tools in mathematics.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.718. The natural logarithm is particularly important in calculus and complex analysis because of its useful properties in mathematical modeling and solving real-world problems.
One key characteristic of natural logarithms is that \( \ln e = 1 \). This property is used extensively since it simplifies expressions involving \(e\). For instance, in our problem, after applying properties of logarithms, the expression \( \ln(e^{-3x}) \) simplifies to \(-3x \ln e\). Recognizing that \( \ln e = 1 \) allows us to further simplify this to \(-3x\).
The function \( \ln(x) \) is the inverse of the exponential function with base \(e\). This relationship explains why natural logarithms appear frequently in problems involving exponential growth or decay, such as population models or radioactive decay problems. Recognizing these properties helps in seamlessly transforming expressions across different forms, particularly when dealing with exponential equations.

Exponential Functions

Exponential functions have the form \(f(x) = a \cdot e^{bx}\), where \(e\) is Euler's number, \(a\) is a constant, and \(b\) controls the rate of growth or decay. These functions significantly affect various fields, from natural sciences to finance.
A fundamental property of exponential functions is that they grow at a rate proportional to their value. This intrinsic feature makes them crucial in modeling scenarios where change accelerates over time. For instance, compound interest in finance or population dynamics can often be described with exponential functions.

• Exponential Growth: This occurs when \(b > 0\), making the function increase as \(x\) grows.
• Exponential Decay: Occurs when \(b < 0\), causing the function to decrease as \(x\) increases.

In our exercise, the term \(e^{-3x}\) showcases exponential decay, as indicated by the negative exponent.
Understanding exponential functions is crucial not only to solve such expressions but also to appreciate their application in real-world phenomena where rapid growth or decline is observed. Recognizing these functions and their properties allows mathematicians and scientists to effectively model and predict changes in natural and theoretical systems.