Problem 39 - Calculus and its applications

Question

Given \(\ln 4=1.3863\) and \(\ln 5=1.6094,\) find each value. Do not use a calculator. $$ \ln \frac{4}{5} $$

Step-by-Step Solution

Answer

\( \ln \frac{4}{5} = -0.2231 \).

Step 1: Understand the Problem

We need to find the natural logarithm of the fraction \( \frac{4}{5} \). We are given \( \ln 4 = 1.3863 \) and \( \ln 5 = 1.6094 \). We can use properties of logarithms to express \( \ln \frac{4}{5} \) in terms of \( \ln 4 \) and \( \ln 5 \).

Step 2: Apply the Quotient Rule for Logarithms

According to the quotient rule for logarithms, \( \ln \frac{a}{b} = \ln a - \ln b \). Thus, we can write \( \ln \frac{4}{5} = \ln 4 - \ln 5 \).

Step 3: Substitute Given Values

Using the given values \( \ln 4 = 1.3863 \) and \( \ln 5 = 1.6094 \), substitute these into the equation: \( \ln \frac{4}{5} = 1.3863 - 1.6094 \).

Step 4: Perform the Subtraction

Calculate the subtraction: \( 1.3863 - 1.6094 = -0.2231 \).

Key Concepts

Natural logarithm

The natural logarithm, often written as "ln," is a special logarithm that has a base of Euler's number, denoted as \( e \). This constant \( e \) is approximately equal to 2.71828. The natural logarithm has some unique applications, mainly because of its properties in calculus and its appearance in natural growth and decay models. It helps in simplifying expressions and solving exponential and logarithmic equations. As you work with natural logarithms, remember:

• \( \ln e = 1 \) because \( e^1 = e \).
• \( \ln 1 = 0 \) since any number raised to the power of 0 equals 1.

Natural logarithms are particularly useful because they allow us to work in domains involving continuous growth or processes, such as population growth models or compound interest calculations.

Quotient rule

The quotient rule for logarithms is an extremely important tool when dealing with logarithmic expressions of fractions. This rule states that for any positive numbers \( a \) and \( b \), the natural logarithm of a fraction \( \frac{a}{b} \) can be expressed as the difference of the logarithms:

• \( \ln \frac{a}{b} = \ln a - \ln b \)

This property helps simplify the logarithmic calculation without using technology like a calculator, as shown in the original step-by-step solution. This technique is proper when you have known logarithm values for both numerator and denominator, as with \( \ln 4 = 1.3863 \) and \( \ln 5 = 1.6094 \). By applying this rule, transforming complex logarithmic expressions becomes straightforward, enhancing your ability to solve problems efficiently.

Logarithmic properties

Logarithmic properties are fundamental rules that govern the operations of logarithms. They allow us to manipulate and simplify logarithmic expressions, making complex calculations more approachable. Here are some key properties:

• Product Rule: \( \ln(ab) = \ln a + \ln b \)
• Quotient Rule: \( \ln \frac{a}{b} = \ln a - \ln b \)
• Power Rule: \( \ln(a^b) = b \cdot \ln a \)

These properties mean that logarithms turn multiplication into addition, division into subtraction, and exponentiation into multiplication. Recognizing and understanding these properties are crucial for anyone tackling logarithmic problems. They significantly reduce the complexity of equations and aid in both theoretical and practical applications, such as solving for unknowns in exponential equations or evaluating logarithmic scales in sciences.