Problem 46 - Intermediate Algebra

Question

Evaluate each expression without using a calculator. $$ \ln \frac{1}{e^{3}} $$

Step-by-Step Solution

Answer

\( \ln \frac{1}{e^3} = -3 \).

Step 1: Understanding the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. This means \( \ln(a) = x \) implies \( e^x = a \). Our task is to find \( \ln \left( \frac{1}{e^3} \right) \).

Step 2: Use Properties of Logarithms

One property of logarithms is \( \ln(\frac{a}{b}) = \ln(a) - \ln(b) \). In our expression, \( a = 1 \) and \( b = e^3 \). Therefore, we can rewrite the expression as: \( \ln(1) - \ln(e^3) \).

Step 3: Simplify the Expression

We know that \( \ln(1) = 0 \) because \( e^0 = 1 \), and another property is \( \ln(e^x) = x \). Thus, \( \ln(e^3) = 3 \). This simplifies our expression to: \( 0 - 3 = -3 \).

Step 4: Final Answer

Hence, the value of \( \ln \frac{1}{e^3} \) is \(-3\).

Key Concepts

Properties of Logarithms

When working with logarithms, it is crucial to understand their fundamental properties. These properties help us manipulate and simplify logarithmic expressions and solve equations involving logarithms. Let's dive into some key properties:

• Product Property: The logarithm of a product is the sum of the logarithms of its factors. Mathematically, this is expressed as \( \ln(ab) = \ln(a) + \ln(b) \).
• Quotient Property: The logarithm of a quotient is the difference between the logarithm of the numerator and the denominator. This property is expressed as \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
• Power Property: The logarithm of a power is the exponent times the logarithm of the base, expressed as \( \ln(a^b) = b \cdot \ln(a) \).
• Logarithm of One: The natural logarithm of 1 is always zero since \( e^0 = 1 \), so \( \ln(1) = 0 \).

These properties help simplify complex logarithmic expressions, making them easier to evaluate.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves using the properties of logarithms to make the expressions more manageable. This is especially useful when solving equations or evaluating expressions without a calculator.To simplify \( \ln \left( \frac{1}{e^3} \right) \), we can apply the quotient property of logarithms: \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \). Given that \( a = 1 \) and \( b = e^3 \), we rewrite the expression as:\[ \ln(1) - \ln(e^3) \]We know \( \ln(1) = 0 \) from the property of the logarithm of one. For the term \( \ln(e^3) \), we use the power property and understand that \( \ln(e^3) = 3 \cdot \ln(e) = 3 \) since \( \ln(e) = 1 \).Therefore, our simplified expression becomes:\[ 0 - 3 = -3 \]Simplifying logarithmic expressions by leveraging logarithmic properties allows us to calculate values more smoothly without complicated calculations.

Exponential Functions

Exponential functions play an integral role in various mathematical models, especially in growth and decay problems. The base of natural logarithms, the number \( e \), is also foundational in exponential functions.An exponential function is expressed in the form \( f(x) = a \cdot e^{bx} \), which describes a curve that continually increases or decreases. Here \( a \) and \( b \) are constants, with \( e \) being the mathematical constant approximately equal to 2.71828.

• When \( b > 0 \), the function represents exponential growth, where as \( x \) increases, \( f(x) \) rapidly increases.
• Conversely, if \( b < 0 \), the function models exponential decay, meaning as \( x \) increases, \( f(x) \) declines.

In the context of logarithms, understanding exponential functions is vital as logarithms are the inverse operation of exponential functions. This connection makes solving equations involving exponentials and logarithms manageable and allows us to provide solutions effortlessly, such as calculating \( \ln(e^3) = 3 \). Knowing the behavior and properties of exponential functions further enables a deeper understanding of natural logarithms in mathematics.