Problem 34 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Answer: \(x^{2}+2 y\)

Step 1: Identify the logarithmic property

To simplify the given expression, we need to recognize that it has the form of \(\ln{e^a}\). This property says that the natural logarithm of an exponential function is equal to the exponent itself: \(\ln{e^a} = a\).

Step 2: Apply the logarithmic property to the given expression

We have the given expression: $$\ln e^{x^{2}+2 y}$$ Using the logarithmic property from step 1, we can simplify it to: $$x^{2}+2 y$$ Now, the given expression has been simplified without using a calculator. The simplified expression is: $$x^{2}+2 y$$

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \). The mathematical constant \( e \) (approximately 2.71828) is known as Euler's number. It is an irrational number and is fundamental in mathematics, especially in calculus and complex analysis.

Natural logarithms enjoy special properties that make them very useful in simplifying and solving equations. One key property is that \( \ln e^a = a \). This means that the natural logarithm of \( e \) raised to any power \( a \) is simply \( a \). This property is a direct result of the definition of logarithms as the inverse of exponential functions. So, when you see \( \ln e^{x^2 + 2y} \), using this property allows you to skip the complicated calculation and immediately identify the result as \( x^2 + 2y \).

Understanding the natural logarithm is crucial because it is frequently encountered in mathematics, sciences, and engineering. It simplifies many problems and is a fundamental tool for solving exponential growth and decay problems.

Exponential Functions

Exponential functions are those in which the variable appears in the exponent. This form is typically written as \( f(x) = e^x \), where \( e \) is Euler's number. These functions are renowned for their unique properties, particularly their constant growth rate. The function grows exponentially, which means it increases rapidly as \( x \) increases.

An important characteristic of exponential functions is that their derivative is proportional to the function itself. In mathematical terms, \( \frac{d}{dx} e^x = e^x \). This property is unique to exponential functions with base \( e \), making them a critical part of calculus and differential equations.

Exponential functions also play a vital role in modeling real-world scenarios, such as population growth, radioactive decay, and interest calculations in finance. Their natural connection with logarithms, especially the natural logarithm, makes them easier to manipulate and solve when combined with logarithmic equations.

Logarithmic Identities

Logarithmic identities are equations that hold true for logarithms and allow us to manipulate and simplify logarithmic expressions easily. These identities are fundamental tools in mathematics, enabling the transformation of complex logarithmic expressions into simpler forms.

Here are some key identities related to natural logarithms:

• Product Identity: \( \ln(a \cdot b) = \ln a + \ln b \)
• Quotient Identity: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)
• Power Identity: \( \ln(a^b) = b \cdot \ln a \)
• Change of Base Formula: \( \log_b a = \frac{\ln a}{\ln b} \)

In the context of our given expression \( \ln e^{x^2 + 2y} \), we apply the identity \( \ln e^a = a \) to directly simplify the expression to \( x^2 + 2y \). This particular logarithmic property simplifies calculations involving \( e \) exponentially, which is why it is exceptionally important. Understanding these identities can greatly aid in working through logarithmic problems efficiently.