Problem 35 - Precalculus

Question

Evaluate the expression. \(\ln \left(e^{5}\right)\)

Step-by-Step Solution

Answer

The expression simplifies to 5.

Step 1: Identify the Expression

Analyze the expression \(\ln \left(e^{5}\right)\). Identify that it involves the natural logarithm (\(\ln\)) and an exponential expression \(e^{5}\).

Step 2: Apply the Logarithm Rule

Recall the property of logarithms: \(\ln(e^{x}) = x\). This is because the natural logarithm and the exponential function are inverse functions.

Step 3: Simplify the Expression Using the Rule

Apply the rule from Step 2 to simplify \(\ln \left(e^{5}\right)\). Thus, \(\ln \left(e^{5}\right) = 5\).

Step 4: Verify the Result

Verify that the simplification follows logically: since \(\ln\) and \(e\) are inverses, \(\ln(e^{5})\) directly simplifies to 5.

Key Concepts

Exponential Function: Understanding "e"

In mathematics, an exponential function is one that involves the constant "e," a special irrational number approximately equal to 2.71828. It's used extensively in mathematics to model growth and decay processes.

The most basic type of exponential function is written as \( e^x \), where "e" is the base and "x" is the exponent. This function grows very rapidly as the value of \( x \) increases, which makes it very powerful in calculations that require rapid growth modeling. The function \( e^x \) is particularly unique because it is its own derivative, meaning the function's rate of change is always proportional to its current value.

When evaluating expressions involving \( e \), you typically see it in contexts where growth, decay, or compounding interest is described. The base \( e \) becomes vital for understanding natural logarithms, which are logarithms to the base "e." These are used to "reverse" the exponential operation in calculus and algebra.

Logarithm Rules: Simplifying Complex Expressions

Logarithms are the inverse operations of exponentiation and can effectively "undo" exponential functions. A logarithm can be thought of as asking the question: "To what power must we raise the base to get a certain number?" For example, the natural logarithm \( \ln \) uses the base \( e \), so \( \ln(e^x) = x \). This formulation is an essential rule because it highlights how the logarithm and exponential functions are related.

Some critical logarithm properties simplify complex expressions include:

• Product Rule: \( \ln(xy) = \ln(x) + \ln(y) \)
• Quotient Rule: \( \ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y) \)
• Power Rule: \( \ln(x^c) = c \cdot \ln(x) \)

These rules allow us to break down and evaluate logarithmic expressions efficiently, revealing the internal relationships between exponential and logarithmic functions.

Specifically for natural logarithms, the rule \( \ln(e^x) = x \) holds because of the cancellation effect between the logarithm and its inverse, the exponential function.

Inverse Functions: The Relationship between Logarithms and Exponentials

Inverse functions are pairs of functions that "undo" each other. If you apply a function and then its inverse, you will return to your original value. This is an important concept as it creates a bridge between different mathematical operations.

The natural logarithm \( \ln(x) \) and the exponential function \( e^x \) are classic examples of inverse functions. For any real number \( x \), applying \( \ln \) and then \( e \), or vice versa, brings you back to that original \( x \). In equation form, this is expressed as:

• \( \ln(e^x) = x \)
• \( e^{\ln(x)} = x \)

This relationship is crucial for solving equations where the variable is an exponent. It also makes simplifying expressions like \( \ln(e^5) \) straightforward since you immediately deduce it to be \( 5 \), as the functions \( \ln \) and \( e^x \) cancel out each other's operations.

Understanding inverse functions and their properties is vital in higher-level mathematics, including calculus, because they provide insight into how mathematical models and computations can be simplified and interpreted.