Problem 33 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln 2 x y $$

Step-by-Step Solution

Answer

The expression can be rewritten as \(\ln 2 + \ln x + \ln y\).

Step 1: Recognize the Product Inside the Logarithm

First, notice the expression inside the logarithm, \(2xy\), is a product of two values: 2, x, and y

Step 2: Apply the Product Rule of Logarithms

The product rule of logarithms states that the logarithm of a product can be expressed as the sum of the logarithms of its factors. Thus the expression can be split into \(\ln 2 + \ln x + \ln y\).

Step 3: Simplify the Expression

We simplify the expression. Since the natural logarithm of 2, \(\ln 2\), is a constant, the expression simplifies to \[\ln 2 + \ln x + \ln y\].

Key Concepts

Product Rule of Logarithms

When working with logarithms, one of the most useful properties is the product rule. This rule helps simplify expressions where a logarithm encompasses a product of terms. Here's the simple principle: the logarithm of a product is equal to the sum of the logarithms of each factor. For example, consider the expression \( \ln(2xy) \). Here, we're dealing with the natural logarithm of a product—specifically, the numbers 2, \( x \), and \( y \).
By applying the product rule, we can break this down into the sum \( \ln 2 + \ln x + \ln y \). This makes it much easier to handle and manipulate logarithmic expressions.

• This property is incredibly useful for solving equations involving logarithms, where you can simplify and isolate terms.
• It’s also a powerful tool to transform complex logarithmic forms into simpler, more intuitive ones.

Mastering this rule will enable you to effectively tackle more advanced mathematical problems involving logarithms.

Natural Logarithms

Natural logarithms are a special type of logarithm, denoted as \( \ln \), which operate on the base \( e \). The number \( e \) is an irrational mathematical constant approximately equal to 2.71828. Encountering natural logarithms in various scientific and engineering disciplines is common, owing to their connection with growth processes and continuous compounding.
Why are natural logarithms important? The function \( \ln(x) \) helps model real-world situations where rates change continuously, such as population growth or interest calculations in finance.

• Natural logarithms facilitate calculations involving exponential growth and decay.
• They simplify solving continuous systems due to base \( e \) being a fundamental mathematical constant.

Understanding how to manipulate \( \ln \) will make it easier for you to work with models and equations both theoretical and applied contexts.
Keep in mind that the same logarithmic properties apply to natural logs too, including our previously discussed product rule.

Simplification of Logarithmic Expressions

Simplifying logarithmic expressions is a valuable skill, especially when handling complex equations. The aim is to use logarithmic properties to transform a cumbersome expression into a more manageable form. For instance, starting with \( \ln(2xy) \), you can use the product rule of logarithms to break it down into \( \ln 2 + \ln x + \ln y \). This step by step simplification:

• Makes it easier to understand the expression by isolating each component.
• Allows further algebraic operations or solving for \( x \) without getting lost in the complexity.

It's crucial to recognize when and how to apply properties like the product, quotient, or power rules to achieve a simplified form.
As you become more practiced with these techniques, you'll find that working with logarithms becomes less intimidating and more intuitive.
By consistently simplifying, we avoid mistakes and prepare expressions for more straightforward integration into larger problems or equations.