Problem 28 - Calculus of a Single Variable

Question

In Exercises 19–28, use the properties of logarithms to expand the logarithmic expression. $$ \ln \frac{1}{e} $$

Step-by-Step Solution

Answer

-1

Step 1: Understand what 'expand' means

To 'expand' a logarithmic expression, simply means to rewrite it in a way that no longer contains the operation of taking a logarithm of a product, quotient, or power. In this case, we are looking to expand \(\ln \frac{1}{e}\).

Step 2: Apply the Quotient Rule

By the Quotient Rule, \(\ln \frac{a}{b}\) can be rewritten as \(\ln a - \ln b\). Applying this rule gives \(\ln 1 - \ln e\).

Step 3: Apply natural logarithm rules

Two things to recall here are that the natural logarithm of 1 is zero, \(\ln 1 = 0\), and the natural logarithm of \(e\) is one, \(\ln e = 1\). Applying these rules, we get \(0 - 1\).

Key Concepts

Properties of Logarithms

When dealing with logarithmic expressions, understanding their properties is crucial. Logarithms, in general, have several properties that simplify complex expressions. Among them, you can find rules for handling products, quotients, powers, and even specific values like zero or the base of the logarithm.

• The Product Rule: For \(\log_b (m\cdot n) = \log_b m + \log_b n\).
• The Quotient Rule: Enables us to write \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\).
• The Power Rule: States that \(\log_b (m^n) = n \cdot \log_b m\).

Each of these properties helps transform and simplify logarithmic expressions, making calculations more manageable. Applying these rules will lead to accurate and efficient solutions.

Quotient Rule of Logarithms

The Quotient Rule of logarithms is essential when you're dealing with logarithms of fractions. This property states that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator. In math terms, \(\log_b \left(\frac{a}{b}\right) = \log_b a - \log_b b\).
This becomes handy when simplifying expressions that look complex but aren't difficult once you apply this rule. For example, the expression \(\ln \left(\frac{1}{e}\right)\) uses the quotient rule to rewrite it as \(\ln 1 - \ln e\).
Remember that this rule only works directly with logarithms, making it a consistently useful tool in your math toolkit. It allows the transformation of division in logarithms into subtraction, simplifying the entire process.

Natural Logarithm Properties

Natural logarithms, which are based on the constant \(e\), have unique properties that simplify calculations. The natural logarithm is denoted as \(\ln\) and possesses all standard logarithmic properties with additional insights concerning the number \(e\).

• \ \ln e = 1: This states that the natural logarithm of its own base is always 1.
• \ \ln 1 = 0: This asserts that any number raised to zero equals 1, hence its log is zero as well.

In the step-by-step solution, these properties significantly simplify the expression \(\ln \left(\frac{1}{e}\right)\). By recognizing that \(\ln 1 = 0\) and \(\ln e = 1\), you quickly arrive at \(0 - 1\), a straightforward result.

Logarithmic Expansion

Logarithmic expansion involves breaking down a logarithm into simpler parts using its core properties. The process involves employing the product, quotient, and power rules to deconstruct a given expression.
By expanding logarithmic expressions, we aim to eliminate any complexities such as products or quotients within the logarithm. Instead, they are rewritten as sums or differences of simpler logarithms.
In the exercise, the expansion of \(\ln \left(\frac{1}{e}\right)\) through these principles allows it to be expressed as \(\ln 1 - \ln e\). Then, using the known natural log values for 1 and \(e\), the expression simplifies to \(0 - 1\). As such, expansion not only makes the expression easier but also highlights the value of understanding logarithmic properties thoroughly.