Problem 39 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

\( \ln e^{2/3} = 2/3 \).

Step 1: Understand the Logarithmic Identity

The natural logarithm, denoted as \( \ln \), has the property that \( \ln e^x = x \). This is because the natural logarithm \( \ln \) is the inverse function of the exponential function \( e^x \).

Step 2: Apply the Identity to the Given Expression

We can directly apply the identity from Step 1 to the given expression \( \ln e^{2/3} \). According to the identity, since \( x = 2/3 \), \( \ln e^{2/3} = 2/3 \).

Key Concepts

Exponential Function

An exponential function is a mathematical function of the form \( f(x) = a^x \), where "a" is a positive constant known as the base. In most cases, when dealing with natural logarithms, the base is the irrational number \( e \) (approximately equal to 2.71828). This specific case is known as the natural exponential function, expressed as \( f(x) = e^x \).

• This function is widely used in modeling growth processes, such as populations, financial investments, and certain types of decay, like radioactive decay.
• The base \( e \) is unique because it has the property that the rate of growth of the function matches the current value of the function.

The exponential function \( e^x \) is crucial in many areas of mathematics and its applications because it captures continuous growth. Knowing about this function helps us understand the behavior of natural logarithms, since they are directly related.

Logarithmic Identity

A logarithmic identity helps us simplify expressions involving logarithms, particularly making use of the fact that logarithms are the inverses of exponentials. One essential identity to remember is \( \ln e^x = x \).

• This identity shows that when you take the natural logarithm of \( e \) raised to any power, the result is just the exponent itself, effectively 'undoing' the exponential operation.
• It utilizes the core relationship between logarithms and exponential functions, emphasizing their inverse nature.

Understanding this identity not only simplifies complex expressions but also helps solve equations where the variable is in an exponent. In the context of the problem \( \ln e^{2/3} \), the identity clarifies that the result is simply \( 2/3 \).

Inverse Function

An inverse function essentially reverses the effect of the original function. If a function \( f(x) \) takes an input \( x \) and gives an output \( y \), then its inverse \( f^{-1}(x) \) takes \( y \) as input and gives \( x \) as output.

• The natural logarithm \( \ln(x) \) and the exponential function \( e^x \) are inverse functions.
• This means for any real number \( x \), \( \ln(e^x) = x \) and \( e^{\ln x} = x \).

The concept of inverse functions is pivotal in mathematics because it allows us to reverse processes and solve equations involving exponentials. In daily applications, understanding inverse functions helps decipher patterns and predictions in various fields, such as calculus, physics, and finance.