Problem 22 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression without using a calculator. $$\ln \frac{1}{e}$$

Step-by-Step Solution

Answer

-1

Step 1: Understanding the natural logarithm

Firstly, it's crucial to understand that \(\ln\), the natural logarithm, is the base-\(e\) logarithm where \(e\) is a mathematical constant approximately equal to 2.71828. This natural logarithm works as the inverse of the exponential function, meaning \(\ln(e^a)=a\).

Step 2: Applying the property of natural logarithm

Recall that the expression \(\frac{1}{e}\) can be rewritten as \(e^{-1}\). Therefore, we can rewrite the original equation as \(\ln(e^{-1})\). Now, applying the property of natural logarithm, this expression simplifies to \(-1\).

Step 3: Final expression

After applying the property of natural logarithm, we find that the expression \(\ln \frac{1}{e}\) simplifies to \(-1\). This is the final answer.

Key Concepts

Exponential Function

The exponential function is a powerful mathematical concept that involves the constant base \(e\), approximately equal to 2.71828. This constant is very important in mathematics because it appears frequently in many areas, such as calculus and complex numbers. When we talk about an exponential function, we generally mean a function of the form \(f(x) = e^x\). Here, \(x\) is the exponent to which the constant \(e\) is raised.

Exponential functions have certain key properties:

• Growth pattern: They grow rapidly as the input increases, which is why they are often used to model growth processes, such as populations or investments.
• Continuous and smooth: The graph of an exponential function is a continuous curve that has no breaks, making it very smooth.

Understanding the connection between the exponential function and the natural logarithm is essential, as it helps solve equations where \(x\) is in the exponent. In these situations, logarithms come into play to help isolate \(x\).

Inverse Function

Inverse functions are mathematical operations that "undo" each other. If you have a function \(f(x)\) and its inverse \(f^{-1}(x)\), applying \(f\) followed by \(f^{-1}\) will yield the original input \(x\). This process can be likened to a lock and a key fitting perfectly.

In the context of exponential functions, the inverse function is the natural logarithm. Specifically, if we work with the exponential function \(e^x\), the natural logarithm \(\ln(x)\) is its inverse. This means that if you take an exponential expression \(e^a\) and apply \(\ln\) to it, the result is just \(a\), because \(\ln(e^a) = a\). This inverse relationship is foundational in solving many mathematical problems, particularly those involving growth or decay rates.

Understanding inverse functions can simplify complex problems, especially in calculus, where you often need to "reverse" operations to find derivatives or integrals.

Logarithmic Properties

Logarithms are an important concept in mathematics, providing a method to work with large numbers more easily by converting them into manageable scales. Logarithmic properties allow us to simplify expressions that would otherwise be complex.

Some key properties of logarithms include:

• Product Property: \(\ln(ab) = \ln(a) + \ln(b)\). This shows that the logarithm of a product is the sum of the logarithms of the factors.
• Quotient Property: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\). This indicates that the logarithm of a quotient is the difference of the logarithms.
• Power Property: \(\ln(a^b) = b\ln(a)\). This means if you have an exponent inside the logarithm, you can bring it out front as a multiplier.

These properties are particularly useful in algebra and calculus, as they help in breaking down and simplifying expressions. When dealing with the natural logarithm, these same properties apply, allowing you to solve logarithmic equations with confidence.