Problem 23 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, apply the inverse properties of logarithmic and exponential functions to simplify the expression. $$ \ln e^{x^{2}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression \( \ln e^{x^{2}} \) is \( x^{2} \).

Step 1: Understanding Given Expression

The given expression here is \( \ln(e^{x^{2}}) \). In this expression, \( e^{x^{2}} \) is the argument of the natural logarithm function, \( \ln \).

Step 2: Apply Logarithmic-Exponential Inverse Property

The inverse property between the natural logarithm, \( \ln \), and the exponential function, \( e^{x} \), states that \( \ln(e^{x}) = x \). Applying this property to the given expression, \( \ln(e^{x^{2}}) \) simplifies to \( x^{2} \).

Key Concepts

Inverse Properties in Logarithmic and Exponential Functions

One of the remarkable features of logarithms and exponential functions is their inverse relationship. Imagine these functions as operations that "undo" each other. This property is quite handy when simplifying expressions.

Consider the natural logarithm, denoted as \( \ln(x) \), and the exponential function, represented as \( e^x \). When you take the natural logarithm of an exponential function, or vice versa, the result naturally simplifies. Specifically, \( \ln(e^x) \) simplifies to just \( x \). Why? Because the natural logarithm and the exponential "cancel each other out."

Here’s another interesting aspect:

• \( e^x \) represents growth or decay, familiar in exponential growth models.
• \( \ln(x) \) helps us transform multiplicative relationships into additive ones.

Understanding this inverse relationship is crucial when you're working with complex mathematical expressions. It allows us to transform complicated expressions into much simpler forms with ease.

Exponential Functions

Exponential functions describe processes that grow or decay at a constant rate. They have the form \( f(x) = a \, e^{bx} \), where \( a \) is a constant, \( e \) is the base of the natural logarithm, and \( b \) determines the growth or decay factor.

The base \( e \), approximately 2.718, is unique because it simplifies the mathematics of continuous growth. Think about how interest compounds in finance or how populations grow. These real-world phenomena are often modeled using exponential functions.

Here are some key points:

• Growth: When \( b > 0 \), the function represents growth, and the graph rises rapidly.
• Decay: When \( b < 0 \), it represents decay, with the graph approaching zero.

Exponential functions are powerful tools in various fields, from biology to finance, making them a fundamental concept in understanding how changes occur over time.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \). It is essential for dealing with exponential growth and decay problems, thanks to its deep connections with derivative and integral calculus.

Unlike other logarithmic functions, the natural logarithm offers special properties:

• Derivative: The derivative of \( \ln(x) \) is \( \frac{1}{x} \), a key result in calculus.
• Integration: It arises naturally when integrating \( \frac{1}{x} \).

Consider this: the function \( \ln(x) \) grows slowly. If you plot its graph, you'll notice a gradual increase as \( x \) grows bigger. This makes it useful for understanding slow progressive changes, such as half-life in radioactive decay or pH levels in chemistry.

Mastering the natural logarithm gives you a deeper insight into the math behind everyday changes and scientific problems.