Problem 83 - Intermediate Algebra

Question

Write each logarithmic expression as one logarithm. See Example 7. $$ \ln \left(\frac{x}{z}+x\right)-\ln \left(\frac{y}{z}+y\right) $$

Step-by-Step Solution

Answer

\( \ln \left( \frac{\frac{x}{z} + x}{\frac{y}{z} + y} \right) \)

Step 1: Understand the Problem

We have two separate natural logarithm expressions that involve division, and our goal is to combine them into a single logarithmic expression.

Step 2: Recall the Logarithmic Property

Use the property of logarithms: \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \). This will allow us to combine the logarithms into one.

Step 3: Apply the Property

Apply the property to combine the logarithms: \( \ln \left( \frac{x}{z} + x \right) - \ln \left( \frac{y}{z} + y \right) = \ln \left( \frac{\frac{x}{z} + x}{\frac{y}{z} + y} \right) \).

Step 4: Simplify the Expression

The expression \( \ln \left( \frac{\frac{x}{z} + x}{\frac{y}{z} + y} \right) \) is our simplified, single logarithm version of the original expression. Leave it in this form unless further simplification is required by the context of your problem.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. In essence, it is used to describe the time needed for continuous growth to reach a certain level. Natural logarithms are commonly found in calculus and natural sciences due to their relationship with exponential functions.

• The key to understanding \(\ln(x)\) lies in recognizing that it's the inverse of the exponential function \(e^x\).
• This means that \(\ln(e^x) = x\) and \(e^{\ln(x)} = x\) as long as \(x > 0\).
• Natural logarithms bear an innate relationship to various phenomena in nature and mathematics, which makes them extremely useful.

Understanding how the natural logarithm behaves allows students to handle equations involving exponential growth or decay, compute compound interest, and solve problems involving continuous growth processes effectively.

Logarithmic Properties

Logarithmic properties are a set of rules that simplify the combination and manipulation of logarithmic expressions. These properties help streamline expressions and are crucial for solving logarithmic equations.

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

Applying these properties allows us to rewrite multiple logarithmic terms into simpler forms or combine them into a single logarithm. For example, in the original exercise, the Quotient Property enables the transformation of the expression with two logarithm terms into one. Understanding and applying these rules effectively play a key role in simplifying logarithmic expressions and solving related problems efficiently.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is an important skill that allows you to make complex logarithmic equations more manageable. Here’s how you can simplify expressions using different properties.

Combining Using Logarithmic Properties

Simplification often involves combining multiple logs into a single term using properties such as the Product, Quotient, and Power properties. This approach reduces the complexity of expressions and facilitates easier problem solving.

• In the exercise, we used the Quotient Property: \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \).
• This allowed the transformation of \( \ln\left(\frac{x}{z} + x\right) - \ln\left(\frac{y}{z} + y\right) \) into a streamlined form: \( \ln\left(\frac{\frac{x}{z} + x}{\frac{y}{z} + y}\right) \).

By mastering these transformations, students can handle logarithmic problems with greater ease. They provide a solid foundation for tackling complex logarithmic equations and can be extended to exponential and related mathematical problems.