Problem 97 - College Algebra

Question

Evaluate or simplify each expression without using a calculator. $$ e^{\ln 5 x^{2}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(e^{\ln 5 x^{2}}\) is \(5x^2\).

Step 1: Understand the Problem and Plan the Solution

The goal is to simplify \(e^{\ln 5 x^{2}}\). Consider \(5 x^{2}\) as a single term and use the property of natural logarithm and exponential function to simplify the expression.

Step 2: Apply the property of Exponential and Natural Logarithms

Apply the property that when the exponential function with base \(e\) is composed with the natural logarithm, they cancel each other out. Hence, \(e^{\ln{5x^2}} = 5x^2.\)

Step 3: Final Result

The resulting simplified form of the expression \(e^{\ln 5 x^{2}}\) is \(5x^2\).

Key Concepts

Natural Logarithmic Properties

The natural logarithm, denoted as \( \ln \), is a logarithm with the base of the constant \( e \), which is approximately 2.718. Natural logarithms are used extensively in mathematics and science to represent exponential growth and decay in a variety of contexts.

One crucial property of natural logarithms is how they interact with exponential functions. Specifically, the property \( e^{\ln x} = x \) holds true for any positive \( x \). This property illustrates that the exponential function \( e^x \) and the natural logarithm \( \ln x \) are inverse functions of each other.

This inverse relationship is key when simplifying expressions such as \( e^{\ln 5 x^{2}} \). By recognizing that applying \( \ln \) and raising it as an exponent of \( e \) effectively "undoes" each other, the expression simplifies directly to the argument of the \( \ln \), which is \( 5x^2 \).

Remember this interaction whenever dealing with these functions; it simplifies seemingly complex expressions greatly.

Exponential Functions

Exponential functions are a category of functions where a constant base is raised to a variable exponent, typically expressed as \( y = a^x \). The base \( a \) is a constant, and one of the most important exponential functions used is when \( a = e \), known as the natural exponential function, denoted as \( y = e^x \).

Exponential functions with base \( e \) have unique properties that make them incredibly important for continuous growth and compound interest calculations. They can model real-world phenomena such as population growth, radioactive decay, and temperature change.

Due to their unique growth rate—a feature called "continuous growth"—exponential functions with base \( e \) are common in solving differential equations and other advanced mathematical models. They grow quickly and continuously, making them ideal for representing processes that accelerate swiftly over time.

Understanding these functions' behavior helps in manipulating and simplifying expressions, especially when coupled with operations involving natural logs, as seen in the expression \( e^{\ln 5x^2} \).

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and arithmetic operations. They form the backbone of algebra and involve manipulation to simplify or solve equations.

When working with algebraic expressions, you'll often perform operations like addition, subtraction, multiplication, and division on both constants and variables. Additionally, you may need to apply rules and properties for exponents and logarithms—such as knowing \( e^{\ln x} = x \) —to simplify these expressions.

Consider the expression \( e^{\ln 5x^2} \). Here, \( 5x^2 \) acts as a single term within the natural logarithm. Understanding how to handle the exponent \( 2 \) across the entire term is crucial; it signifies \( x \) being squared.

Properly managing variables and constants within algebraic expressions allows you to simplify and solve problems efficiently. By observing how logarithmic and exponential properties can simplify calculations, students can solve complex expressions with ease. The importance of these skills extends into calculus and other higher-level mathematics where such manipulations are routine.