Problem 35 - Contemporary Precalculus

Question

Evaluate the given expression without using a calculator. $$e^{\ln x^{2}}$$

Step-by-Step Solution

Answer

Question: Evaluate the expression $$e^{\ln x^{2}}$$ without using a calculator. Answer: The simplified expression is $$x^{2}$$.

Step 1: Recall properties of exponential and logarithmic functions

We will use the properties of exponential functions and logarithms to simplify the expression. In particular, we will use the property: $$e^{\ln a} = a$$ for any positive real number a.

Step 2: Apply the properties of exponential and logarithmic functions

Using the property from step 1, we can rewrite the given expression as follows: $$e^{\ln x^{2}} = x^{2}$$ We have simplified the given expression, and the final answer is: $$x^{2}$$

Key Concepts

Logarithmic Functions

Logarithmic functions are essential in understanding the behavior of exponential equations. They help to 'undo' exponentiation, serving as the inverse of exponential functions.
In any equation where you see a logarithm, it tells you the power to which you must raise a base in order to achieve a given number. The natural logarithm, denoted as \( \ln(x) \), is particularly important because it uses the number \( e \) (approximately 2.718) as its base.
Knowing the basic property that \( e^{\ln a} = a \) for any positive \( a \) is crucial. This property allows for the simplification seen in our original problem: converting \( e^{\ln x^2} \) directly to \( x^2 \). This direct conversion is because \( e \) and \( \ln \) are inverse operations and thus cancel each other out in this context.
Using logarithms effectively can help simplify complex expressions, improve problem-solving speed, and provide deeper insights into the structure of algebraic problems involving exponential relationships.

Properties of Exponents

The properties of exponents are the rules that guide how we handle expressions with powers. These rules include things like multiplying powers (adding their exponents) or raising a power by another power (multiplying the exponents):

• The product of powers rule: \( a^m \cdot a^n = a^{m+n} \)
• The power of a power rule: \( (a^m)^n = a^{m \cdot n} \)
• The power of a product rule: \( (ab)^n = a^n \cdot b^n \)

These rules are interconnected with logarithmic properties because logarithms deal with the power that a certain base number is raised to, which makes understanding exponent rules even more valuable.
In exponential and logarithmic functions, such as in our exercise, recognizing and applying these properties correctly can simplify complex expressions—just like transforming \( e^{\ln x^2} \) to \( x^2 \) using the inverse relationship between exponential and logarithmic functions.
An effective understanding of these exponent properties ensures we handle transformations accurately and confidently, simplifying the process of evaluating expressions.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form without changing their value. This process often requires the use of exponent and logarithm properties.
The goal of simplification is to make expressions easier to work with or to identify equivalent expressions, just as seen in our example \( e^{\ln x^2} = x^2 \). Here, we reduced the expression using a known property, creating a straightforward form that is simpler to understand and apply in further calculations.
Important strategies for simplifying include:

• Identifying common factors or terms
• Canceling out reciprocal terms
• Utilizing properties of logarithms and exponents effectively to condense the expressions

Simplification isn't just about reducing the number of terms or operations; it's about improving clarity and functionality in mathematical expression. With practice, these techniques become valuable tools, making more complex algebraic and calculus problems much more manageable.