Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression.\(\ln e^{x^{2}}\)

Step-by-Step Solution

Answer

\(\ln e^{x^{2}}\) simplifies to \(x^{2}\)

Step 1: Understanding the Inverse Property of Logarithm

The Inverse Property of logarithms states that \(log_b(a^m) = m * log_b(a)\) where 'm' is any real number. This property is very useful for simplifying logs since it allows one to pull the exponent out of the log and multiply it by the log itself.

Step 2: Apply the Inverse Property to the given expression

Here, \(log_b(a^m)\) corresponds to \(\ln e^{x^{2}}\). In this scenario, 'e' is the base, 'x^2' is the exponent. The inverse property allows us to simplify as 'x^2' * \(\ln e\). As \(\ln e = 1\), the \(\ln e^{x^{2}}\) simplifies to \(x^{2}\)

Step 3: Final output

So, the simplified form of \(\ln e^{x^{2}}\) using Inverse Property of Logarithm is \(x^{2}\)

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They are essential in the world of mathematics because they allow us to work with exponential growth or decay in a more manageable way. A logarithm, expressed as \( \log_b(x) \), essentially answers the question: 'To what power must we raise the base \( b \) to get \( x \)?'

The natural logarithm, denoted as \( \ln(x) \), is a special type of logarithm where the base is the mathematical constant \( e \), which is approximately equal to 2.71828. When you see an expression like \( \ln e^{x^{2}} \), it's essentially asking for the power to which \( e \) must be raised to result in \( e^{x^{2}} \), which, according to the Inverse Property of Logarithms, simplifies directly to \( x^{2} \).

Exponential Functions

Exponential functions are powerful tools for modeling growth and decay processes, such as population growth, radioactive decay, and compound interest. The function is typically written as \( f(x) = a^{x} \), where \( a \) is the base and \( x \) is the exponent. The base \( e \) of the natural exponential function is a fundamental constant and has numerous applications in calculus and complex analysis.

In the context of simplifying logarithmic expressions like \( \ln e^{x^{2}} \), recognizing the interplay between exponential and logarithmic functions helps to simplify the expression significantly. By understanding that \( e \) and \( \ln \) are inverse operations, the expression condenses, revealing the clear and simplified result.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. The goal is to transform a complex expression into a more easily understandable form without changing its value. This often involves using properties of operations, such as the distributive property, combining like terms, or, as in the case of \( \ln e^{x^{2}} \), employing the Inverse Property of logarithms.

Simply put, when you encounter a logarithmic function applied to an exponential with the same base, you can apply the Inverse Property to drastically simplify the expression. In this instance, the expression \( \ln e^{x^{2}} \) simplifies to \( x^{2} \), which is much more straightforward to comprehend and utilize in further calculations or applications.

Properties of Logarithms

Understanding the properties of logarithms can significantly ease the process of simplifying complex logarithmic expressions. These properties include the Power Rule, Product Rule, Quotient Rule, and the Change of Base formula. Crucial to our simplification process is the Inverse Property, which states that \( \log_b(b^{x}) = x \) for any base \( b \), except when \( b <= 0 \). This property is particularly useful when dealing with natural logarithms, \( \ln(x) \), and the natural base, \( e \).

When you come across an expression like \( \ln e^{x^{2}} \), the Inverse Property tells us that since the logarithm's base and the exponential's base are the same, they 'cancel out,' leaving us with just the exponent, \( x^{2} \). This is a perfect demonstration of how properties of logarithms make complex expressions more manageable and comprehensible.