Problem 88 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of natural logarithms to rewrite the expression. $$\ln \frac{1}{e^{4}}$$

Step-by-Step Solution

Answer

The given expression \( \ln \frac{1}{e^{4}} \) simplifies to be -4.

Step 1: Recognizing the Natural Logarithm Rule

Given the expression \( \ln \frac{1}{e^{4}} \), you need to recognize that logarithms of divisions can be written as the subtraction of logarithms. In other words, \( \ln(\frac{a}{b}) = \ln(a) - \ln(b) \). Also, the natural logarithm of e equals 1: \(\ln(e) = 1\).

Step 2: Apply the Logarithm Division Rule

Now apply the rule. We have: \( \ln 1 - \ln e^{4} \).

Step 3: Further Simplification

Note that \(\ln{1} = 0\) and that the logarithm of an exponent can be reduced to a multiplication, \(\ln a^b = b \ln a\). We have: \( 0 - 4 \ln e = -4 \) since \(\ln e = 1\).

Key Concepts

Natural Logarithm Rules

Understanding the properties of the natural logarithm is essential when handling logarithmic expressions. One of the fundamental concepts to grasp is that the natural logarithm, denoted as \(\ln(x)\), is the inverse of the exponential function where the base is the mathematical constant \(e\). Key rules of natural logarithms include:

• The natural logarithm of 1 is always 0: \(\ln(1) = 0\).
• The natural logarithm of \(e\) to any power \(x\) gives you that power: \(\ln(e^x) = x\).
• The natural logarithm of a product \(ab\) can be written as the sum of the logarithms of \(a\) and \(b\): \(\ln(ab) = \ln(a) + \ln(b)\).
• The inverse rule states that \(e^{\ln x} = x\) and \(\ln(e^x) = x\), a property frequently used to simplify logarithmic expressions.

These rules are the building blocks that allow students of mathematics to simplify and solve logarithmic expressions with ease. By internalizing these properties, students can quickly recognize how to handle a variety of logarithmic problems.

Logarithm Division Rule

A key operation in logarithms is division, which has its own specific rule known as the logarithm division rule. This rule is critical when simplifying logarithms that involve fractions. The rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator:
\[\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\]
This rule can be seen as an extension of the product rule since it reflects the inverse relationship between multiplication and division in logarithmic terms. It is this principle that allows us to take a complex logarithmic expression involving division and break it down into more manageable parts. In the context of the natural logarithm, applying this rule simplifies calculations considerably because it transforms the division inside the logarithm into a straightforward subtraction problem outside of it.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves applying a series of rules and properties of logarithms to reduce a complex expression to its simplest form. When tackled step by step, simplification can become a routine process. To simplify expressions like \(\ln \frac{1}{e^{4}}\), we don't just apply the division rule but also utilize the exponents rule and the fundamental truths about \(\ln(1)\) and \(\ln(e)\). The exponent rule states that the logarithm of a number raised to an exponent is the exponent times the logarithm of the number:
\[\ln(a^b) = b \cdot \ln(a)\]
Incorporating these principles, simplifying logarithmic expressions becomes an exercise in detection and substitution. Students often struggle with keeping track of the numerous properties, but with practice, recognizing which rules to apply becomes almost second nature. The process often entails converting divisions into subtractions, turning products into additions, and simplifying powers to their coefficients, leading to a much more straightforward numeric or algebraic solution.