Problem 36 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Answer: The simplified form of the given expression is \(\ln 2\).

Step 1: Recall Properties of Logarithms and Exponentials

Before we begin simplifying the expression, let's recall some properties of logarithms and exponentials: 1. The base of the natural logarithm, denoted as \(\ln\), is the Euler's number (\(e\)). 2. \(\ln(a^b) = b \cdot \ln(a)\). 3. \(e^{\ln(a)} = a\). Now that we have these properties in mind, let's move on to simplifying the given expression.

Step 2: Simplify the Expression Inside the Exponential Function

Given the expression: $$e^{\ln (\ln 2)}$$ We know that \(e^{\ln(a)}=a\), which means that if we have an expression of the form \(e^{\ln(a)}\), we can just replace it with \(a\). In this case, our \(a\) is \((\ln 2)\), so the expression becomes: $$e^{\ln (\ln 2)} = \ln 2$$

Step 3: Final Simplification

We have simplified the expression to: $$\ln 2$$ This is already in its simplest form, and since we're not using a calculator, we don't need to evaluate the natural logarithm any further. So, the final answer is: $$e^{\ln (\ln 2)} = \ln 2$$

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is fundamental for simplifying exponential expressions. Logarithms are a way of expressing exponentiation in reverse. Here are a few key properties:

• Power Rule: \( \ln(a^b) = b \cdot \ln(a) \). This means the logarithm of a power translates the exponent as a multiplier outside the logarithm.
• Logarithms and Exponentials: If you have something in the form \( e^{\ln(a)} \), you can simplify it to \( a \). This property indicates that an exponential and a logarithm with the same base cancel each other out due to their inverse relationship.

These properties allow us to solve complex expressions by breaking them down into simpler terms that are easier to handle.
They provide a structured method for transforming and evaluating logarithmic and exponential forms.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a logarithm to the base \( e \), where \( e \) is approximately 2.71828. This special logarithm has many useful properties:

• It is used to solve equations where the base is \( e \), simplifying the complexity involved with continuous growth or decay.
• It is particularly beneficial in calculus because its derivative is 1, making differentiation straightforward.

Because \( e \) is the natural base, the natural logarithm appears frequently in growth models, economics, biology, and even in financial calculations.
It serves as a bridge to understanding the relationship between simple and complex growth processes.

Euler's Number

Euler's number, denoted as \( e \), is a mathematical constant that is the base of the natural logarithm. Its value is approximately 2.71828. This number is fundamental:

• It appears in various contexts across mathematics, especially in calculus, where it describes exponential growth and decay.
• \( e \) is known for the unique property that the function \( f(x) = e^x \) has the same rate of increase as its value at any given point; that is, its derivative \( f'(x) = e^x \) is \( e^x \).
• It is an irrational number, meaning it cannot be expressed as a simple fraction, similar to \( \pi \).

Euler's number is essential in mathematical models that include continuous growth, finance for compounding interest, and even in the field of complex numbers when addressing equations involving growth rates.