Problem 42 - Calculus Early Transcendentals

Question

Logarithm properties Use the integral definition of the natural logarithm to prove that \(\ln (x / y)=\ln x-\ln y\)

Step-by-Step Solution

Answer

Question: Use the integral definition of the natural logarithm to prove the logarithm property \(\ln(x/y) = \ln x - \ln y\). Answer: We used the integral definition of the natural logarithm, \(\ln x = \int_1^x \frac{1}{t} dt\), to prove the logarithm property by following these steps: 1. Define \(z = \frac{x}{y}\) and rewrite the integral definition for \(\ln z\): \(\ln z = \int_1^z \frac{1}{t} dt\). 2. Substitute \(v = yt\), rewrite the integral, and split it into two parts: \(\ln z = \frac{1}{y} \left( \int_y^1 \frac{1}{v} dv + \int_1^x \frac{1}{v} dv \right)\). 3. Rewrite the integrals as natural logarithms: \(\ln z = \frac{1}{y} (\ln \frac{1}{y} + \ln x)\). 4. Simplify using the property \(\ln \frac{1}{y} = -\ln y\): \(\ln z = \ln x - \ln y\). Thus, we have successfully proven that \(\ln(x/y) = \ln x - \ln y\).

Step 1: Recall the integral definition of the natural logarithm

The integral definition of the natural logarithm of a positive number \(x\) is given by: $$\ln x = \int_1^x \frac{1}{t} dt$$

Step 2: Define variables

Let's define \(z = \frac{x}{y}\). Then our goal is to show that \(\ln z = \ln x - \ln y\).

Step 3: Write the integral definition for \(\ln z\)

Using the integral definition, we can write \(\ln z\) as: $$\ln z = \int_1^z \frac{1}{t} dt$$

Step 4: Rewrite the integral using substitution

Recall that we defined \(z = \frac{x}{y}\). We can rewrite the integral as follows: $$\ln z = \int_1^{\frac{x}{y}} \frac{1}{t} dt$$ Now let's perform a substitution: let \(v = yt\). Then, \(t = \frac{v}{y}\), and consequently \(dt = \frac{1}{y} dv\). Replacing the limits of integration, when \(t=1\), \(v=y\), and when \(t=\frac{x}{y}\), \(v=x\). Now we can rewrite the integral as: $$\ln z = \int_y^x \frac{1}{v} \frac{1}{y} dv$$

Step 5: Split the integral into two parts

Now we can split the integral into two parts: $$\ln z = \int_y^x \frac{1}{v} \frac{1}{y} dv = \frac{1}{y} \int_y^x \frac{1}{v} dv$$ Now we'll split the integral into two integrals: $$\frac{1}{y} \int_y^x \frac{1}{v} dv = \frac{1}{y} \left( \int_y^1 \frac{1}{v} dv + \int_1^x \frac{1}{v} dv \right)$$

Step 6: Rewrite the integrals as natural logarithms

Now we can rewrite the two integrals as natural logarithms: $$\frac{1}{y} \left( \int_y^1 \frac{1}{v} dv + \int_1^x \frac{1}{v} dv \right) = \frac{1}{y} (\ln \frac{1}{y} + \ln x)$$

Step 7: Simplify and obtain the result

Using the property that \(\ln \frac{1}{y} = -\ln y\), we can simplify the expression: $$\frac{1}{y} (\ln \frac{1}{y} + \ln x) = \frac{1}{y} (-\ln y + \ln x) = \ln x - \ln y$$ So we have proven that \(\ln (x / y) = \ln x - \ln y\).

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln x \), is a logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. It is particularly useful in calculus and mathematical analysis. The natural logarithm of a number \( x \) is the power to which \( e \) must be raised to obtain the number \( x \).
This function is defined for all positive real numbers and has many applications in growth processes, science, and engineering, due to its connection with exponential growth models. Understanding the integral definition of the natural logarithm helps in developing a deeper intuition about its properties and applications.

Integral Definition

The integral definition of the natural logarithm is a fundamental concept that connects logarithms to calculus. It states:

\[\ln x = \int_1^x \frac{1}{t} \, dt\]

This formula means that the natural logarithm of \( x \) is equal to the area under the curve of \( \frac{1}{t} \) from 1 to \( x \).
Understanding this integral definition is essential because it demonstrates how the concept of the natural logarithm can be derived from the area under a curve. It's a powerful method for proving various logarithmic identities, such as \( \ln (x / y) = \ln x - \ln y \), by using known calculus techniques.

Substitution Method

The substitution method is a technique often used in calculus to simplify complex integrals. It involves replacing a variable in the integral with another variable that makes integrating easier.

In the context of logarithms, consider proving \( \ln (x / y) = \ln x - \ln y \). First, set \( z = \frac{x}{y} \). Then, apply substitution in the integral formula for \( \ln z \). By letting \( v = yt \), translating limits, and considering how \( t \) and \( v \) relate, you can simplify the integration process.
This step strategically converts the problem into one involving simpler integrals and paves the way to demonstrate logarithmic properties efficiently.

Properties of Logarithms

The properties of logarithms are crucial for simplifying and solving logarithmic expressions. In this problem, we focus on one key property:

• \( \ln (a/b) = \ln a - \ln b \) - The logarithm of a quotient is the difference of logarithms.

These properties simplify calculations in a wide range of mathematical problems. They help break down complex expressions into manageable parts. Using integral concepts and transformation techniques, these properties can often be proved or derived, as shown in this exercise, helping students understand the underlying principles beyond simple memorization.