Problem 51 - Intermediate Algebra

Question

Evaluate each expression without using a calculator. $$ \ln e^{-7} $$

Step-by-Step Solution

Answer

The expression evaluates to -7.

Step 1: Recognize the Properties of Logarithms and Exponents

The natural logarithm function, \( \ln \), is the inverse of the exponential function with base \( e \). This means that \( \ln(e^x) = x \). Recognizing this property will help simplify the expression.

Step 2: Rewrite the Expression Using Logarithm Properties

Given the expression \( \ln e^{-7} \), use the property from Step 1. By rewriting \( e^{-7} \) as \( e^{(-7)} \), you can directly apply the logarithm property: \( \ln(e^{-7}) = -7 \).

Step 3: Simplify the Expression

By applying the property \( \ln(e^x) = x \), from the rewritten form \( \ln(e^{-7}) = -7 \), the simplified value of the expression is \(-7\).

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a special type of logarithm that has the constant \( e \) as its base. The number \( e \) is approximately 2.718 and is an important constant in mathematics, particularly in the realms of calculus and exponential growth.
The natural logarithm has a unique relationship with the exponential function because it serves as its inverse.
This means that the natural logarithm of \( e \) raised to any power \( x \), written as \( \ln(e^x) \), simply simplifies to \( x \).
The reason this simplification works is that the logarithm is asking: "to what power must \( e \) be raised to get \( e^x \)?" The answer is clearly \( x \) itself.
Understanding these concepts can make it easier to work with expressions involving \( \ln \), helping solve them without a calculator.

Core Properties of Logarithms

Logarithms, including the natural logarithm, follow specific properties that assist in simplifying expressions and solving equations:

• **Product Property:** \( \log_b(MN) = \log_b M + \log_b N \).
• **Quotient Property:** \( \log_b(\frac{M}{N}) = \log_b M - \log_b N \).
• **Power Property:** \( \log_b(M^n) = n \cdot \log_b M \).

In the context of natural logarithms, these properties help solve more complex expressions without a calculator. For the given exercise, the **Power Property** was especially useful.
By employing the idea that \( \ln(e^{-7}) = -7 \), the exercise showcases the simplicity that can come with understanding these rules.
Learning to apply these properties systematically will make handling any logarithmic or exponential expression much more approachable.

Exploring the Exponential Function

An exponential function is any function where a constant base is raised to a variable exponent.
One of the most widely used bases is \( e \), the basis of the natural logarithm, leading to expressions of the form \( e^x \).
The inverse of an exponential function is a logarithmic function. This relationship is crucial in mathematics because it allows for solving equations where the variable is in the exponent.
This exercise uses the fact that \( \ln(e^x) = x \) to pinpoint the value of \( \ln(e^{-7}) \) as \(-7\).
Becoming comfortable with exponential functions not only aids in algebra, but it is paramount in understanding growth processes, chemical reactions, and many other natural phenomena where change accelerates at a consistent percentage.