Problem 43 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression evaluates to 6.

Step 1: Recall the Property of Logarithms

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \). A key property of logarithms is that if you have \( \ln(e^x) \), the result is simply \( x \). This is derived from the definition of a logarithm, which states that \( \ln(e^x) = x \).

Step 2: Apply the Property to the Given Expression

Given the expression \( \ln(e^6) \), use the property from Step 1. According to the property \( \ln(e^x) = x \), this simplifies directly to 6.

Key Concepts

Properties of Logarithms

Logarithms have several properties that make them powerful in simplifying complex calculations. Understanding these properties can help you manipulate and simplify expressions involving logarithms and exponentials. One of the most fundamental properties is:

• The Logarithmic Identity: For any positive number \( a \), the equation \( \ln(e^a) = a \) holds true. This is because the logarithm base \( e \) and the exponential function base \( e \) are inverse operations. So, they effectively 'cancel' each other out, leaving the exponent.

Moreover, there are other properties like the Product, Quotient, and Power Rules:

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

Understanding these allows you to fix complicated logarithms into simpler parts and solve them easily.

Exponential Functions

Exponential functions are mathematical functions of the form \( f(x) = a^x \), where \( a \) is a positive constant and \( x \) is the exponent. One of the most important exponential functions is \( e^x \), where \( e \approx 2.718 \, (a \text{ mathematical constant}) \). These functions have several key characteristics:

• Rapid Growth: Exponential functions increase quickly as \( x \) becomes larger.
• Inverse Functions: The exponential function \( e^x \) is the inverse of the natural logarithm \( \ln x \).
• Continuous and Differentiable: These properties make exponential functions a central part in calculus and differential equations.
• Applications: Exponential functions model real-world phenomena like radioactive decay and population growth.

Understanding exponential functions helps in comprehending their relationship with logarithms and in solving various types of equations.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They are expressed as \( f(x) = \log_b(x) \) or in the case of natural logs, \( f(x) = \ln(x) \) with base \( e \). Logarithmic functions have unique features and understandings that are essential for solving equations and modeling certain types of data.Here are some key points:

• Inverse Properties: Because they are inverse functions, \( \ln(e^x) = x \) and \( e^{\ln x} = x \).
• Logarithmic Scales: These scales fix the issue of large numbers, using logarithms to condense data and make patterns easier to see.
• Applications: Logarithmic functions are used in measuring sound intensity in decibels and in determining pH in chemistry.
• Mathematical Operations: With its properties, it provides a handle on complex data present in scientific studies.

By mastering logarithmic functions, you gain a powerful tool to solve real-world problems involving exponential growth and decay.