Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression without using a calculator.\(\ln \frac{1}{e^{3}}\)

Step-by-Step Solution

Answer

-3

Step 1: Rewrite the Division as Multiplication

We know that division by a number is the same as multiplying by its reciprocal. Therefore, we can rewrite the argument of the logarithm as: \(\frac{1}{e^{3}} = e^{-3}\). So, \(\ln \frac{1}{e^{3}}\) becomes \(\ln {e^{-3}}\).

Step 2: Apply properties of logarithms

A property of logarithms states that \(\ln a^{b} = b \ln a\). Applying this property to the argument of the logarithm, we find that \(\ln {e^{-3}} = -3 \ln e\).

Step 3: Simplify the expression

In the base \(e\) logarithm, \(\ln e = 1\). So in the expression, -3 * \(\ln e\) simplifies to -3.

Key Concepts

Properties of Logarithms

Logarithms are powerful tools in mathematics that help us handle large numbers and exponential relationships with ease. One of the key properties of logarithms is that they can transform multiplication into addition and powers into multiplication. This is particularly useful when solving complex equations.

Some important properties to keep in mind include:

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n). \)
• Quotient Rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n). \)
• Power Rule: \( \log_b(m^n) = n\log_b(m). \)

In the original exercise, we used the Power Rule for simplifying the expression. By recognizing that \(\ln a^b = b \ln a\), we could easily transform the logarithm of \(e^{-3}\) into \(-3 \ln e\).

Understanding and applying these properties of logarithms allows us to tackle problems more efficiently by breaking them down into simpler components.

Exponents

Exponents allow us to express repeated multiplication concisely. They have specific rules that govern their usage, and understanding these rules helps simplify many mathematical expressions.

Some key properties of exponents are:

• Multiplication of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Division of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \( (a^m)^n = a^{mn} \)
• Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)

In our exercise, we dealt with a negative exponent. The expression \(\frac{1}{e^3}\) was rewritten using the negative exponent property: \( e^{-3} \). This transformation is often useful because it simplifies the computation, particularly in expressions involving logarithms.

Mastering exponents is essential for dealing with everything from basic algebra to more advanced calculus problems.

Natural Logarithms

Natural logarithms are a specific type of logarithm, with the base being the irrational number \(e \approx 2.71828\). Denoted as \(\ln\), they are crucial in calculus and natural growth processes, like population growth or radioactive decay.

Some key points about natural logarithms include:

• The constant \(e\) is uniquely significant because its natural logarithm equals 1: \( \ln e = 1 \).
• Natural logarithms have all the properties of general logarithms, such as the power, product, and quotient rules.

In our exercise, the simplification of \(\ln e^{-3} = -3 \ln e \) was completed by recalling that \( \ln e = 1 \). This knowledge allowed us to finalize our work by recognizing that \( -3 \ln e\) simplifies directly to \( -3 \).

Understanding how natural logarithms function and their properties enables us to solve exponential equations easily and apply them to real-world contexts.