Problem 17 - Precalculus with Calculus Previews

Question

In Problems \(13-18\), find the exact value of the given logarithm. $$ \ln e^{e} $$

Step-by-Step Solution

Answer

The exact value of \( \ln e^{e} \) is \( e \).

Step 1: Understanding the Logarithm Property

The problem given is to find the value of \( \ln e^{e} \). Begin by recalling the logarithm property: \( \ln a^b = b\ln a \). Here, \(a=e\) and \(b=e\).

Step 2: Applying the Logarithmic Identity

Using the property from the previous step, apply it to the expression \( \ln e^{e} \). This gives us \( e \times \ln e \).

Step 3: Evaluation of Natural Logarithm

Remember that the natural logarithm \( \ln e = 1 \). This is because the natural logarithm of the base \(e\) is 1.

Step 4: Final Calculation

Substitute \( \ln e = 1 \) into the expression \( e \times \ln e \). This simplifies to \( e \times 1 = e \).

Key Concepts

Understanding Logarithm Properties

Logarithms have several key properties that make them powerful tools for solving exponential equations. One of the most important properties is the "Power Rule," which states that for any logarithm of the form \( \ln a^b \), you can rewrite it as \( b \cdot \ln a \). This property allows you to simplify complex logarithmic expressions into more manageable terms.

Let's consider what this means for our specific problem: \( \ln e^{e} \). Here, the base \( a \) is \( e \) and the exponent \( b \) is also \( e \). By applying the Power Rule, the expression \( \ln e^{e} \) transforms into \( e \cdot \ln e \).

Understanding these properties can vastly simplify the calculation process, and allow you to see the relationship between exponents and logarithms more clearly. Properties like this are foundational not only in mathematics but also in fields of science and engineering where exponential growth or decay is analyzed.

Introduction to Exponential Functions

Exponential functions are those where a constant base is raised to a variable exponent, like \( a^x \). The base \( e \), approximately 2.718, is particularly significant because it's the base of natural logarithms. Functions with this base appear in many real-world scenarios, including population growth and radioactive decay.

The function \( e^x \) is special because it is its own derivative and integral, meaning it grows at a rate proportional to its current value—this is unique to the exponential function with base \( e \).

• High growth rates
• Basis for continuous compounding interest
• Natural decay rates and half-lives follow \( e \)

These characteristics make exponential functions crucial for understanding natural processes and changes over time. In our problem, recognizing that \( e^{e} \) retains e as both base and exponent highlights the intrinsic relationship between \( e \) and natural growth processes.

The Process of Evaluating Logarithms

Evaluating logarithms is often about recognizing core identities and applying them correctly. For instance, one of the simplest identities is that \( \ln e = 1 \). This identity is derived from the fact that the natural log of \( e \) asks the question: "To what power must \( e \) be raised to get \( e \)?"—the answer, of course, is 1.

When evaluating expressions like \( \ln e^{e} \), the task becomes a straightforward exercise of applying the Power Rule and known identities. In our example, we started with \( \ln e^{e} \), transformed the expression using the Power Rule to \( e \cdot \ln e \), and then utilized the identity that \( \ln e = 1 \), leading us to the final answer \( e \).

• Identify the logarithm's base
• Apply any relevant properties
• Use known logarithm identities

This systematic approach makes sure that you can solve logarithmic equations confidently and quickly.