Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The simplified form of the expression is \(x^{2}\)

Step 1: Identify the expression

We start with the expression \(e^{ \ln x^{2} }\)

Step 2: Use the property of natural exponentials and logarithms

The relationship between the exponential and logarithm is used to simplify the expression. The property that we are going to use is \(e^{ \ln a } = a\), where \(a\) is any positive number.

Step 3: Apply this property to the expression

Applying this property to our expression, we replace \(a\) with \(x^{2}\) to get \(e^{ \ln x^{2} } = x^{2}\)

Key Concepts

Properties of Logarithms

Logarithms possess unique properties that make them powerful tools for simplifying complex expressions. One such property is the inverse relationship between logarithms and exponentials. This relationship is essential because it allows us to interchangeably use these functions to solve equations, especially those involving natural exponential functions like 'e'.
Another vital property of logarithms is the power rule: \(\cdot \ln(a) = \ln(a^n)\). This means you can move the exponent in the logarithm to the front as a multiplier. For example, \(\ln(x^2) = 2 \cdot \ln(x)\). This rule can make simplifying logarithmic expressions much easier, especially when combined with other properties.

• Product Rule: \(\ln(ab) = \ln(a) + \ln(b)\)
• Quotient Rule: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)

Understanding these properties helps in simplifying expressions involving logarithms and exponents.

Natural Logarithms

The natural logarithm is denoted as \(\ln\) and is the logarithm to the base 'e', where 'e' is an irrational number approximately equal to 2.71828. Natural logarithms are common in calculus and are often used in solving problems involving growth and decay because of their well-loved properties.

The primary property of natural logarithms is their inverse relationship with the exponential function \(e^x\). Specifically, for any positive number \(a\), the equation \(e^{\ln a} = a\) holds true. This means if you take the natural logarithm of a number, then use it as an exponent for 'e', you return to your original number.

• Useful in calculus, especially for solving integrals and derivatives
• Simplifies the process of solving exponential equations

Understanding natural logarithms is essential for dealing with exponential functions, simplifying expressions, and solving complex equations.

Simplifying Expressions

In mathematics, simplifying expressions means reducing them to their simplest form, making them easier to work with or understand. This process is crucial in problem-solving as it helps you eliminate complexity and find solutions efficiently.

When you have an expression like \(e^{ \ln x^{2} }\), you can use the previously mentioned properties to simplify it directly to \(x^2\). Here are some general tips for simplifying expressions:

• Use co-function identities to rewrite expressions for easier handling.
• Substitute functions with their simplified forms if applicable.
• Always check if expressions can be reduced using fundamental properties of logarithms or exponentials.

Simplification not only makes calculations more straightforward but also aids in understanding the underlying structure of the problem.