Problem 46 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Logarithmic Expression

The expression given is \( \ln \frac{1}{e^{3}} \). The "ln" represents the natural logarithm, which is the logarithm to the base \( e \). Our goal is to simplify this expression to find its exact value.

Step 2: Applying the Logarithmic Rule for Fractions

Recall that \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \). Here, we can apply this rule to our expression: \( \ln \frac{1}{e^3} = \ln 1 - \ln e^3 \).

Step 3: Evaluating the Logarithm of 1

The natural logarithm of 1 is always 0, i.e., \( \ln 1 = 0 \). This simplifies our expression to \( 0 - \ln e^3 \).

Step 4: Applying the Logarithmic Power Rule

Use the power rule of logarithms, which states that \( \ln a^b = b \cdot \ln a \). Thus, \( \ln e^3 = 3 \cdot \ln e \). Since \( \ln e = 1 \), we have \( 3 \cdot 1 = 3 \).

Step 5: Final Simplification

Substitute back into the expression \( 0 - \ln e^3 \), which simplifies to \( 0 - 3 = -3 \).

Key Concepts

Understanding Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve logarithms, which are the inverse operations of exponentiation. In the given exercise, the expression \( \ln \frac{1}{e^3} \) features the natural logarithm, denoted as "ln". The natural logarithm uses the base \( e \), where \( e \approx 2.71828 \).

The focus of logarithmic expressions is to decipher the power to which a base must be raised to yield a particular number. For example, in this expression, we're asked to determine the power that \( e \) should be raised to, in order for \( \frac{1}{e^3} \) to be the resulting value. Tackling expressions like these requires understanding how to manipulate both the base and the logarithmic rules effectively. A solid grasp of these concepts lays the groundwork for simplifying and solving complex logarithmic expressions.

Key Logarithmic Rules

Logarithmic rules are the fundamental guidelines that help in manipulating and simplifying logarithmic expressions. In the given problem, a few key rules are applied:

• Product Rule: \( \ln(ab) = \ln a + \ln b \)
• Quotient Rule: \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \)
• Power Rule: \( \ln(a^b) = b \cdot \ln a \)

In the exercise, the Quotient Rule is applied first to break down \( \ln \frac{1}{e^3} \) into more manageable terms: \( \ln 1 - \ln e^3 \). Since \( \ln 1 = 0 \), it brings us to the next application of the Power Rule: \( \ln e^3 \) becomes \( 3 \cdot \ln e \). Knowing that \( \ln e = 1 \), we simplify it further to reach the final answer.

Simplifying Expressions with Logs

Simplifying expressions that involve logarithms can be quite straightforward once you understand the rules. Let's break it down using the exercise where the original expression was \( \ln \frac{1}{e^3} \).

The first step involves utilizing the Quotient Rule for logs. This gives \( \ln 1 - \ln e^3 \). We know by definition that \( \ln 1 = 0 \), allowing us to simplify the expression immediately to \( 0 - \ln e^3 \).

Next, the Power Rule helps in simplifying \( \ln e^3 \). By rewriting this as \( 3 \cdot \ln e \), and knowing \( \ln e = 1 \), we simplify it to 3. The expression thus simplifies to \( 0 - 3 \), which equals −3.

The ability to simplify expressions like this relies heavily on knowing how to apply the rules correctly and recognizing constants like \( \ln e \) that can further ease the process.