Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify the given expression. $$ e^{-2 \ln x} $$

Step-by-Step Solution

Answer

The simplified expression is \(1/x^2\).

Step 1: Understanding the Expression

We are given the expression \(e^{-2 \ln x}\). Our goal is to simplify this expression. Recall that \(\ln x\) represents the natural logarithm of \(x\).

Step 2: Apply Logarithmic Identity

We use the identity for logarithms: \(-\ln x = \ln(1/x)\). Therefore, \(-2 \ln x = \ln((1/x)^2) = \ln(1/x^2)\).

Step 3: Convert Using the Exponential Identity

We know that \(e^{\ln a} = a\). Using this property, we have: \(e^{\ln(1/x^2)} = 1/x^2\). Thus, \(e^{-2 \ln x} = e^{\ln(1/x^2)}\).

Step 4: Simplify the Expression

Using the above identities, we simplify: \(e^{-2 \ln x} = 1/x^2\). That is the simplified form of the given expression.

Key Concepts

Exponential Function

When we talk about an exponential function, we're discussing a function of the form \( f(x) = a^x \), where \( a \) is a constant and \( x \) is a variable. The special case where the base \( a \) is Euler's number \( e \) (approximately 2.71828) is particularly significant in mathematics due to its unique properties. The function \( f(x) = e^x \) grows faster than any polynomial function as \( x \) increases.

• It's used to model growth and decay processes.
• It has a constant relative rate of growth, described as \( f'(x) = e^x \).

Understanding the behavior of exponential functions is key in calculus and many applied sciences. They appear naturally when dealing with continuous growth processes, such as population growth or interest calculations in finance.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is the logarithm to the base \( e \). It's the inverse operation of the exponential function \( e^x \). So, if \( y = \ln x \), then \( e^y = x \). Here's why it's essential:

• It simplifies multiplication into addition, a crucial property in calculus.
• It's particularly useful in solving equations involving exponentials.

Natural logarithms are often used in various scientific calculations where base \( e \) naturally arises, such as compounding interests and time-based decay.

Logarithmic Identity

Logarithmic identities can transform complicated expressions into more manageable forms. One useful identity is \(-\ln x = \ln(1/x)\). This identity allows us to flip the base of the logarithm when its sign is negated. For example, \(-2 \ln x = \ln((1/x)^2)\). This manipulation helps to simplify expressions like \( e^{-2 \ln x} \).

• Transforms division into subtraction, another property in the logarithmic world.
• Enables the use of other exponential properties for simplification.

This ability to rewrite expressions is a powerful tool in solving algebraic and calculus problems.

Exponential Identity

The exponential identity \( e^{\ln a} = a \) forms the backbone of simplifying expressions that involve both exponentials and logarithms. Using this identity, you can transform an exponent-logarithm combination into a simpler algebraic expression.
For example, understanding that \( e^{\ln(1/x^2)} = 1/x^2 \), allows a direct simplification of expressions such as \( e^{-2 \ln x} \) into \( 1/x^2 \).

• It shows direct equivalency, bridging the gap between supposed complex and simple terms.
• Aids in solving logarithmic equations through conversion back to original terms.

Comprehending this concept is crucial for mastering topics in calculus and higher mathematics, where exponential and logarithmic functions are frequently encountered together.