Problem 45 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression evaluates to -1.

Step 1: Understand the Properties of Logarithms

Before solving the problem, recall the fundamental properties of logarithms, especially the natural logarithm. We know that the natural logarithm, \( \ln \), satisfies \( \ln(e^x) = x \) and \( \ln(a^b) = b\ln(a) \). These will help us evaluate the expression.

Step 2: Rewrite the Expression

The expression given is \( \ln \frac{1}{e} \). Rewriting \( \frac{1}{e} \) using exponents, we get \( e^{-1} \). Therefore, the expression becomes \( \ln(e^{-1}) \).

Step 3: Apply the Logarithm Power Rule

Using the power rule for logarithms, which states that \( \ln(a^b) = b\ln(a) \), we can simplify \( \ln(e^{-1}) \) to become \( -1 \cdot \ln(e) \).

Step 4: Evaluate \( \ln(e) \)

Remember that by the definition of natural logarithms, \( \ln(e) = 1 \). Therefore, substituting into the expression from Step 3, we have: \(-1 \cdot \ln(e) = -1 \cdot 1 = -1\).

Step 5: State the Final Answer

After simplifying the expression using the properties of logarithms, we find that \( \ln \frac{1}{e} = -1 \).

Key Concepts

Properties of Logarithms

Logarithms, especially the natural logarithm denoted as \( \ln \), have special properties that make complex calculations simpler. Understanding these properties is key to simplifying expressions. One of the fundamental properties is the identity \( \ln(e^x) = x \). This tells us that taking the natural logarithm of \( e \) raised to any power yields that power.

Another important property is the product rule, \( \ln(ab) = \ln(a) + \ln(b) \), and the quotient rule, \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \). These rules help in breaking down complex logarithmic expressions into simpler terms. In particular, the latter will be useful when tackling fractions in logarithmic form.

Exponent Rules

Exponential properties are essential when dealing with bases like \( e \) in logarithmic expressions. One crucial exponent rule is that \( a^{-b} = \frac{1}{a^b} \). This rule allows us to convert expressions like \( \frac{1}{e} \) into exponential form, \( e^{-1} \).

Another important rule is that any number raised to the power of zero equals one, \( a^0 = 1 \). These rules help when modifying expressions to apply further simplifications either using logarithms or basic arithmetic operations. Knowing how to move between fraction notation and exponent notation makes evaluating logarithms much more straightforward.

Logarithm Power Rule

The logarithm power rule is a cornerstone when simplifying logarithmic expressions involving exponents. It states that \( \ln(a^b) = b \cdot \ln(a) \). This rule essentially allows you to "bring the exponent down" as a multiplier in front of a logarithm.

In our exercise, this rule is used when simplifying \( \ln(e^{-1}) \). Applying the power rule gives \(-1 \cdot \ln(e) \), which brings us a step closer to simplifying the expression entirely. Since \( \ln(e) = 1 \), the calculation simplifies to \(-1 \cdot 1 = -1\). Understanding this rule helps reduce complex exponential-log expressions into simple arithmetic evaluations.

Simplifying Logarithmic Expressions

The process of simplifying logarithmic expressions often involves using a combination of different properties and rules. Initially, rewriting the given expression into a simpler equivalent forms the basis of simplifying. For instance, converting \( \ln(\frac{1}{e}) \) to \( \ln(e^{-1}) \) was crucial in applying the power rule.

This also goes hand-in-hand with evaluating known values of the logarithm. Since \( \ln(e) = 1 \), any expression involving \( \ln(e) \) quickly simplifies as needed. These strategies form a systematic approach to solving logarithmic problems, making them less daunting and more accessible for learners.