Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( \frac{e^x}{x} \).

Step 1: Review the Expression

We start with the expression \( e^{x - \ln x} \). The base of the exponential function here is \( e \), and the exponent is \( x - \ln x \).

Step 2: Apply Properties of Logarithms and Exponents

Recall the property of exponents: \( a^{m-n} = \frac{a^m}{a^n} \). We can separate the exponent \( x - \ln x \) into two parts, resulting in: \( e^{x - \ln x} = e^x \cdot e^{-\ln x} \).

Step 3: Simplify the Logarithmic Component

The term \( e^{-\ln x} \) can be simplified using the principle that \( e^{\ln a} = a \) if \( a > 0 \). Hence, \( e^{-\ln x} = \frac{1}{e^{\ln x}} = \frac{1}{x} \).

Step 4: Combine Simplified Terms

Now, combine the terms we have: \( e^{x - \ln x} = e^x \cdot \frac{1}{x} = \frac{e^x}{x} \).

Step 5: Final Expression

The simplified expression is \( \frac{e^x}{x} \).

Key Concepts

Exponential Functions

Exponential functions are a cornerstone of calculus and mathematical modeling, often represented in the form \( f(x) = a^x \) where \( a \) is a positive constant known as the base, and \( x \) is the exponent. In this exercise, the exponential function uses the mathematical constant \( e \) as its base. This constant, approximately equal to 2.718, is key in many growth and decay processes found in natural and finance environments.
When dealing with exponential functions, it's crucial to understand their growth rate. Such functions can grow rapidly, depending on the value of the exponent. They also have unique properties that allow them to be manipulated in ways other standard functions cannot, such as the principal property \( e^x \), where its derivative is also \( e^x \), making it invaluable in calculus for solving differential equations.
For example, in \( e^{x - \ln x} \), we employ the property of exponents in step-by-step transformations that shed light on how these powerful functions can be simplified and manipulated, ultimately leading to a more refined expression.

Logarithms

Logarithms are the inverse operation to exponentiation, meaning they allow you to determine what exponent you need to raise a base to get another number. A logarithm with base \( e \) is called a natural logarithm, denoted as \( \ln x \). These are particularly useful for solving equations where the unknown variable is an exponent.
Key properties of logarithms include:

• \( \ln(ab) = \ln a + \ln b \)
• \( \ln(a^b) = b \cdot \ln a \)
• \( \ln(1/a) = -\ln a \)

In the expression from our exercise, \( e^{\ln x} = x \) is a fundamental principle used in simplifying \( e^{-\ln x} \). This turns into \( \frac{1}{x} \), showcasing the power of logarithms to transform equations and simplify complex expressions. Without understanding logarithms, expressions with exponential terms would be much harder to work through.
Once you comprehend logarithms, you unlock a toolkit that lets you handle a variety of transformations and simplifications in mathematical expressions, whether solving equations, modeling real-world behaviors, or working through calculus problems.

Mathematical Simplification

Mathematical simplification involves reducing an expression or equation to its simplest form. This can make equations easier to solve and better to understand. Simplifying might involve factoring, canceling terms, or using mathematical properties like the ones discussed in the context of exponential and logarithmic functions.
In this problem, the original expression \( e^{x - \ln x} \) is simplified using properties of exponents and logarithms. The expression undergoes a transformation:

• First, using \( a^{m-n} = \frac{a^m}{a^n} \) property, it is rewritten as \( e^x \cdot e^{-\ln x} \).
• Next, the logarithmic property \( e^{\ln x} = x \) helps simplify \( e^{-\ln x} \) to \( \frac{1}{x} \).
• Finally, the entire expression simplifies to \( \frac{e^x}{x} \).

This process of dissecting and breaking down expressions step by step, leveraging various mathematical properties, enables one to simplify and solve otherwise complex problems. Hence, mathematical simplification is a critical skill in all branches of mathematics, enhancing clarity and understanding.