Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$\ln e^{2 / 3}$$

Step-by-Step Solution

Answer

The expression \( \ln e^{2/3} \) evaluates to \( \frac{2}{3} \).

Step 1: Understanding the Expression

The expression given is \( \ln e^{2/3} \). This involves the natural logarithm (\( \ln \)) and the exponential function. The natural logarithm of a number is the power to which \( e \) (Euler's number) must be raised to obtain that number.

Step 2: Applying the Logarithmic Identity

We can use the identity \( \ln e^x = x \) for any real number \( x \). This identity tells us that the natural logarithm and exponential functions are inverses of each other.

Step 3: Simplifying the Given Expression

Given the identity \( \ln e^x = x \), apply it directly to the expression \( \ln e^{2/3} \). Based on the identity, this simplifies directly to \( \frac{2}{3} \).

Key Concepts

Exponential Functions

Exponential functions are a fundamental concept in mathematics, often expressed in the form \( f(x) = a^x \), where \( a \) is a constant and \( x \) is a variable. When \( a \) is the mathematical constant \( e \) (approximately 2.718), the function becomes \( e^x \), known as the natural exponential function.
Exponential functions describe situations where growth or decay occurs at a constant rate, such as in population growth, radioactive decay, and interest calculations.
Key properties of exponential functions include:

• They are always positive and never touch the x-axis.
• Their rate of change increases or decreases exponentially.

Understanding exponential functions is crucial because many natural processes are modeled using them, and they often show up in different areas of science and finance.

Logarithmic Identities

Logarithmic identities are important tools for simplifying expressions involving logarithms. Logarithms are the inverses of exponential functions, helping to solve equations where the unknown is an exponent.
One of the most useful logarithmic identities is \( \ln e^x = x \). This identity highlights that the logarithm function \( \ln \) (natural log) essentially 'undoes' an exponential function with base \( e \).
Key logarithmic properties include:

• \( \ln 1 = 0 \), because any number raised to the power of 0 is 1.
• \( \ln ab = \ln a + \ln b \), showing that the log of a product is the sum of the logs.
• \( \ln \left(a^b\right) = b \cdot \ln a \), which is particularly useful for simplifying powers inside logs.

These identities not only help in solving logarithmic equations but also make complex calculations manageable. Knowing these can ease understanding of compound interest, biological growth models, and signal decay in physics.

Inverse Functions

Inverse functions are functions that reverse the effect of the original function. For a function \( f(x) \) to have an inverse, it must be one-to-one (bijective), meaning each output is uniquely paired with an input.
The concept of inverse functions is crucial in mathematics as it allows us to "go back" from the outputs of a function to its inputs.
For example:

• If \( f(x) = y \), then the inverse \( f^{-1}(y) = x \).
• The original function and its inverse cancel each other out, meaning \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

The relationship between natural logarithms and exponential functions, as seen in the identity \( \ln e^x = x \), exemplifies inverse functions. Here, \( \ln \) is the inverse of \( e^x \), demonstrating that applying \( \ln \) to an exponentiated \( e \) value simply results in the exponent itself. Recognizing and understanding inverse functions often simplifies complex real-world problems, enabling easier manipulation of equations and functions.