Problem 34 - Calculus and its applications

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find the natural logarithm of the fraction \( \frac{5}{4} \). We are given the natural logarithms of 4 and 5, specifically \( \ln 4 = 1.3863 \) and \( \ln 5 = 1.6094 \).

Step 2: Use Properties of Logarithms

Recall the logarithmic property \( \ln \frac{a}{b} = \ln a - \ln b \). We can use this to compute \( \ln \frac{5}{4} = \ln 5 - \ln 4 \).

Step 3: Substitute Known Values

Substitute the known logarithmic values into the equation: \( \ln \frac{5}{4} = \ln 5 - \ln 4 = 1.6094 - 1.3863 \).

Step 4: Perform the Calculation

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

Step 5: Conclude the Solution

Conclude that the value of \( \ln \frac{5}{4} \) based on our calculations is \( 0.2231 \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a fundamental mathematical function used to relate exponential and logarithmic expressions. It is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. This base \( e \) is particularly important in mathematics due to its unique properties, especially in calculus and complex analysis.

Natural logarithms are widely used in solving exponential growth and decay problems, such as in biology, economics, and physics. They transform multiplicative relationships into additive ones, making them easier to handle. When you see \( \ln x \), think of it as the power to which the base \( e \) must be raised to produce the number \( x \).

For example, if \( \ln 4 = 1.3863 \) it means \( e^{1.3863} \approx 4 \). This transforms exponential equations into more manageable forms.

Properties of Logarithms

Logarithms have several useful properties that make them extremely helpful in simplifying complex problems. These properties allow you to accurately solve logarithmic equations, convert between logarithmic and exponential forms, and even combine or break apart logarithmic expressions.

Some important properties include:

• Product Property: \( \ln(ab) = \ln a + \ln b \)
• Quotient Property: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)
• Power Property: \( \ln(a^n) = n\ln a \)

The problem we are exploring takes advantage of the quotient property. It shows how a more complicated-looking term \( \ln \frac{5}{4} \) can be broken down into simpler known values \( \ln 5 \) and \( \ln 4 \).

By understanding these properties, you can transform expressions and solve them without complex calculations, often simplifying what might initially seem difficult.

Logarithmic Subtraction

Logarithmic subtraction involves using the properties of logarithms to find the difference between two logarithmic values. This is particularly useful when dealing with the division of numbers inside a logarithm.

The subtraction property given by \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \) is a powerful tool. It allows you to evaluate the logarithm of a fraction without needing to compute the fraction directly. Instead, you find the natural logarithms of the individual components and subtract them.

In our original exercise, we used known values, \( \ln 5 = 1.6094 \) and \( \ln 4 = 1.3863 \), to compute \( \ln \frac{5}{4} \). By applying logarithmic subtraction, we find \( \ln \frac{5}{4} = \ln 5 - \ln 4 = 0.2231 \). This method simplifies the process, turning a complex problem into a straightforward calculation.