Problem 50 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the natural logarithmic expression without using a calculator. \(\ln \left(e^{\frac{1}{3}}\right)\)

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{1}{3} \).

Step 1: Identify Components

The expression given is \( \ln \left(e^{\frac{1}{3}}\right) \). Here, we need to evaluate the natural logarithm of an exponential expression. In this case, the base of the logarithm is the natural number \( e \), and the exponent of \( e \) is \( \frac{1}{3} \).

Step 2: Apply the Property of Logarithms

Natural logarithms have a particular property: \( \ln \left( e^x \right) = x \). This is because the natural logarithm is the inverse of the exponential function with base \( e \). In our expression, \( e^x = e^{\frac{1}{3}} \), so \( x = \frac{1}{3} \).

Step 3: Simplify the Expression

Using the property \( \ln(e^x) = x \), we simplify \( \ln \left(e^{\frac{1}{3}}\right) \) to \( \frac{1}{3} \). This simplification is direct because the logarithm and the exponential function cancel each other out.

Key Concepts

Exponential Function

The exponential function is a fundamental mathematical concept where a constant base is raised to a variable exponent. In most cases, the base is the natural number \( e \), approximately equal to 2.71828. When we write \( e^x \), it represents an exponential growth if \( x \) is positive or a decay if \( x \) is negative. The exponential function is particularly interesting because it has a continuous growth rate proportional to its value.
Understanding the exponential function is crucial in various fields like finance, biology, and physics, where growth processes are modeled. For example:

• Population growth in biology often follows an exponential pattern.
• Interest compounding in finance is based on exponential functions.

Grasping this concept helps in deciphering how exponentially changing quantities develop over time. It also builds a foundation for understanding how logarithms can reverse the effect of exponentiation.

Logarithm Properties

Logarithms are the inverse operations of exponentials, much like subtraction is the inverse of addition. The natural logarithm, denoted as \( \ln \), uses the base \( e \). The basic property of logarithms is \( \ln\left(e^x\right) = x \), showing the direct relationship between logarithms and exponential terms.
Some important properties include:

• Product Property: \( \ln(ab) = \ln(a) + \ln(b) \).
• Quotient Property: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
• Power Property: \( \ln(a^b) = b \ln(a) \).

These rules simplify complex logarithmic expressions and enable solving equations involving exponential growth or decay, making logarithms essential in mathematics. They're particularly useful in contexts where results span multiple orders of magnitude.

Inverse Functions

Inverse functions are operations that "undo" each other. For instance, if addition is done by one function, subtraction will be its inverse. In the world of exponential functions and logarithms, the natural logarithm \( \ln \) is the inverse of the exponential function \( e^x \). This means applying these functions successively will cancel each other out, yielding the original input.
For example, if we have an expression \( \ln(e^x) \), it simplifies directly to \( x \) because the logarithm "undoes" the exponentiation. This is a powerful tool in mathematics as:

• It helps solve equations where the variable is in the exponent.
• It makes complex exponential relationships more tractable by turning them into linear forms.

Inverse functions play a crucial role in making sense of how different mathematical operations interact and provide insights into their interconnected nature.