Problem 43 - Intermediate Algebra

Question

Evaluate each expression without using a calculator. $$ \ln e^{6} $$

Step-by-Step Solution

Answer

The value of \( \ln e^6 \) is 6.

Step 1: Understand the Expression

The expression \( \ln e^{6} \) involves the natural logarithm function \( \ln \) and an exponential base \( e \). The goal is to evaluate this expression without a calculator by understanding the properties of logarithms and exponents.

Step 2: Apply the Logarithmic Identity

One of the key properties of logarithms to remember is that the natural logarithm of an exponential function \( \ln(e^x) \) simplifies directly to the exponent itself, which is \( x \). This property holds because of the definition of the natural logarithm as the inverse of the exponential function with base \( e \).

Step 3: Simplify the Expression

Given the logarithmic identity from the previous step, apply it to simplify the expression: \( \ln(e^6) = 6 \). No further simplification is needed.

Key Concepts

Natural Logarithms

Natural logarithms, often abbreviated as "ln," are logarithms with the base of Euler's number, represented by the letter "e." This mathematical constant, approximately 2.718, is fundamental in many areas of mathematics, especially in calculus and mathematical modeling. Natural logarithms are used to solve problems involving growth processes, decay, and continuously compounding interest. Unlike common logarithms, which use a base of 10, natural logarithms use base "e," making them uniquely suited to describe natural phenomena.

• The natural logarithm of a number gives you the time needed to reach that number at a continuous growth rate.
• The relationship between the natural logarithm and the exponential function is a special one, where one undoes the operation of the other—making them inverse functions.
• Understanding this inverse relationship helps simplify complex expressions like ln(e^x).

By recognizing the properties of natural logarithms, such as their ability to simplify expressions involving the exponential function, solving logarithmic problems becomes much easier.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. These functions are written in the form of \( e^x \), where "e" is the constant base and "x" is the exponent. Exponential growth and decay patterns are seen in various scientific and financial contexts, making these functions essential for modeling real-world phenomena.
Exponentiation has a profound impact on the behavior of functions, leading to rapid increases (growth) or decreases (decay) depending on the value and sign of the exponent. Here are some key points:

• An exponential function with base "e" and positive exponent grows continuously and is smooth and continuous.
• The function is defined for all real numbers, meaning it covers both negative and positive values of "x."
• Because of its continuous and smooth nature, the exponential function is an excellent model for continuous growth processes such as compound interest, population growth, and radioactive decay.

These properties make understanding and utilizing exponential functions crucial for solving various mathematical and real-life problems efficiently.

Properties of Logarithms

Logarithms have several fundamental properties that simplify the process of solving logarithmic and exponential equations. These properties are derived from the inverse relationship between logarithms and exponentials, making them powerful tools in mathematical problem-solving. Let's examine some key properties:

• The Product Rule: The logarithm of a product is equal to the sum of the logarithms of the individual numbers: \( \ \ln (ab) = \ln (a) + \ln (b) \).
• The Quotient Rule: The logarithm of a quotient is equal to the difference of the logarithms: \( \ \ln \left( \frac{a}{b} \right) = \ln (a) - \ln (b) \).
• The Power Rule: The logarithm of a power is equal to the exponent times the logarithm of the base: \( \ \ln (a^b) = b \cdot \ln (a) \).
• The Base Rule: When dealing with the base "e," it simplifies any expression: \( \ \ln (e^x) = x \), showing the direct relationship between natural log and exponential functions.

Understanding these properties helps immensely in algebraic manipulation and solving equations, especially when working without calculators. When these properties are applied correctly, they make it easier to simplify expressions like ln(e^x), turning complex logarithmic equations into more manageable ones.