Problem 22 - Calculus of a Single Variable

Question

In Exercises 19–28, use the properties of logarithms to expand the logarithmic expression. $$ \ln (x y z) $$

Step-by-Step Solution

Answer

The expanded form of \(\ln(x y z)\) is \(\ln(x) + \ln(y) + \ln(z)\).

Step 1: Identify the Properties of Logarithms

In this exercise, we are going to use the property of logarithms that states \(\ln(a * b) = \ln(a) + \ln(b)\). This property allows us to take the logarithm of a product (multiple things multiplied together) and transform it into the sum of the logarithms of the individual parts.

Step 2: Apply the Property of Logarithms

Let's apply the property of logarithms to the given expression \(\ln(x y z)\). By the product rule, we can write this as: \(\ln(x) + \ln(y) + \ln(z)\).

Step 3: Write the Final Expanded Form

The final expanded form of the given expression \(\ln(x y z)\) is \(\ln(x) + \ln(y) + \ln(z)\).

Key Concepts

Logarithmic Expressions

Logarithms are math functions you use to solve problems involving exponents and growth patterns. They are simply the inverse operations of exponentiation.
When you see a logarithmic expression, like \( \ln(xyz) \), you're dealing with the natural logarithm (more on that later) applied to a product. You usually see these when simplifying or expanding: you look to change them into simpler forms.
Logarithms can be applied directly to numbers or variables. When they deal with numbers, calculations happen. When they deal with variables, like in our expression, the goal is often to make further operations easier.

• Consider the operation: in \( \ln(xyz) \), \( x \), \( y \), and \( z \) are multiplied together inside the log operation.
• Through properties of logarithms, you can simplify the expression for easier handling.

Product Rule

The product rule is a key property of logarithms. This rule helps you turn a log of a product into a sum of logs. This can be a life-saver for simplifying large or complicated expressions.
Here's how it works: when you have a product inside a logarithm, like \( \ln(a \cdot b) \), you apply the rule to get \( \ln(a) + \ln(b) \). For three variables, like in \( \ln(xyz) \), it becomes \( \ln(x) + \ln(y) + \ln(z) \).
This rule isn't restricted to natural logs alone. It can be used with other bases too, making it quite a versatile tool.

• The product rule helps break down complex expressions into manageable parts.
• It's useful in algebra and calculus wherever you meet multiplying factors inside a logarithm.

Natural Logarithm

The natural logarithm, represented as \( \ln \), is a specific type of logarithm that uses \( e \), a special mathematical constant approximately equal to 2.71828, as its base.
Natural logs simplify calculations in calculus, especially when working with exponential growth and decay phenomena.
You often see natural logarithms in science and engineering due to their efficiency in solving logarithmic problems with base \( e \).

• Natural logs prioritize the constant \( e \) because of its natural connection to continuous growth.
• When you see \( \ln(xyz) \), it lets you instantly recognize the use of constant \( e \) and how the expression would expand using the product rule.