Problem 91 - College Algebra

Question

Evaluate or simplify each expression without using a calculator. $$ \ln \frac{1}{e^{6}} $$

Step-by-Step Solution

Answer

-6

Step 1: Simplify the fraction using logarithmic rules

Observe that the natural logarithm \(ln\) is base \(e\). There's a rule of logarithms that says the logarithm of a base to its exponent is just the exponent. That is, \(ln(a^{k}) = k \cdot ln(a)\). Therefore, when \(a = e\) and \(k = 6\), the expression turns into \(ln(e^{6}) = 6 \cdot ln(e)\). However, in this case it's the reciprocal so it becomes \(-6 \cdot ln(e)\). The expression is now simplified to \(-6 \cdot ln(e)\).

Step 2: Calculate the natural logarithm of \(e\)

By definition, the natural logarithm of \(e\) is 1 (i.e., \(ln(e) = 1\)). So, \(-6 \cdot ln(e)\) becomes \(-6 \cdot 1\).

Step 3: Final calculation

The final step is a simple multiplication: \(-6 \cdot 1\), which results in \(-6\).

Key Concepts

Logarithm Rules

Logarithms have specific rules that help us simplify and evaluate expressions efficiently. The natural logarithm is a type of logarithm with base \( e \), which is approximately equal to 2.718. When working with natural logarithms, it's essential to know these rules so you can manipulate the expressions correctly.

• The Power Rule: The logarithm of a power \( k \) is equal to the exponent times the logarithm of the base. Thus, \( \ln(a^{k}) = k \cdot \ln(a) \). This rule helps reduce complex powers into simpler multiplications involving logarithms.
• The Reciprocal Rule: If you have a fraction \( \frac{1}{a^{k}} \), you can bring the exponent to the front of the logarithm as a negative, \( \ln(\frac{1}{a^{k}}) = -k \cdot \ln(a) \).

Understanding these rules allows you to transform complicated exponential expressions swiftly, as logarithms convert multiplication into addition and powers into products.

Exponents

Exponents are a shorthand way to represent repeated multiplication of the same number. They are fundamental to many aspects of algebra, including logarithms.
The expression \( a^{k} \) means \( a \) multiplied by itself \( k \) times. In expressions like \( \ln(\frac{1}{e^{6}}) \), exponents help define the scale of logarithmic operations.
There are several properties related to exponents that are useful:

• Multiplication of Like Bases: When multiplying like bases, add their exponents. \( a^m \cdot a^n = a^{m+n} \).
• Division of Like Bases: When dividing like bases, subtract the exponents. \( \frac{a^m}{a^n} = a^{m-n} \).
• Negative Exponents: A negative exponent indicates a reciprocal. \( a^{-k} = \frac{1}{a^{k}} \).

Utilizing the rules of exponents allows further simplification within logarithmic contexts, making calculations more straightforward.

Simplifying Expressions

Simplifying expressions can help in evaluating mathematical problems by breaking down complex problems into easier, more manageable parts. When simplifying, you employ various rules from algebra and logarithms to reduce expressions to their most basic form.When dealing with expressions like \( \ln \frac{1}{e^{6}} \), start by using the reciprocal property of exponents, resulting in \( \ln(e^{-6}) \). This follows the power rule, allowing you to express it as \( -6 \cdot \ln(e) \).
The goal of simplification is always to transform expressions into a plain and straightforward format. This involves eliminating complex fractions, consolidating terms, and applying known rules for logarithms and exponents.
Ultimately, simplification bridges the gap between the beginning state of an expression and its fully evaluated solution, providing clarity in mathematical reasoning.

Properties of Logarithms

Logarithms possess unique properties that make them powerful tools in simplifying and solving exponential equations. These properties transform complex operations into simpler tasks, enhancing understanding and efficiency.

• Product Property: The logarithm of a product is the sum of the logarithms of the factors: \( \ln(a \times b) = \ln(a) + \ln(b) \).
• Quotient Property: The logarithm of a quotient is the difference between the logarithms of the numerator and denominator: \( \ln(\frac{a}{b}) = \ln(a) - \ln(b) \).
• Change of Base Formula: Allows conversion between different logarithmic bases: \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \).

These properties play a crucial role in rearranging and simplifying expressions with logarithms. For example, in simplifying \( \ln \frac{1}{e^6} \), you effectively use the quotient or reciprocal properties as part of the main procedure. Understanding these characteristics gives you a robust foundation for tackling various logarithmic problems.