Problem 2 - College Algebra with Modeling and Visualization

Question

Exercises \(I-6:\) (Refer to Example I.) Use a calculator to approximate each pair of expressions. Then state which property of logarithms this calculation illustrates. $$ \ln 12+\ln 5, \quad \ln 60 $$

Step-by-Step Solution

Answer

The expressions illustrate the product property of logarithms: \( \ln(a) + \ln(b) = \ln(ab) \).

Step 1: Understanding the Problem

We're given two logarithmic expressions, \( \ln 12 + \ln 5 \) and \( \ln 60 \), and need to approximate their values using a calculator. Then, we must identify the property of logarithms they illustrate.

Step 2: Calculate Each Expression

First, use a calculator to determine the value of each expression separately. Find \( \ln 12 \approx 2.4849 \) and \( \ln 5 \approx 1.6094 \). Then calculate \( \ln 60 \approx 4.0943 \).

Step 3: Summation of Logarithms

Add the two calculated logarithmic values: \( \ln 12 + \ln 5 \approx 2.4849 + 1.6094 = 4.0943 \).

Step 4: Compare the Results

Notice that the sum of \( \ln 12 + \ln 5 \approx 4.0943 \), which equals \( \ln 60 \approx 4.0943 \). This shows the two expressions are equivalent in value.

Step 5: Identify the Property

The equality of \( \ln 12 + \ln 5 = \ln 60 \) demonstrates the product property of logarithms, which is \( \ln(a) + \ln(b) = \ln(ab) \). Here, \( 12 \times 5 = 60 \), confirming this property.

Key Concepts

Product Property

In logarithms, the product property is a fundamental idea that simplifies complex expressions. It states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this is expressed as:

• \( \ln(a) + \ln(b) = \ln(ab) \)

This property is very useful because it allows us to transform multiplication inside the logarithmic function into addition, which is often easier to calculate and understand. In the exercise example, multiplying 12 and 5 gives 60, which illustrates this property perfectly: \( \ln 12 + \ln 5 = \ln 60 \).
Understanding this property can help simplify many logarithmic expressions and calculations.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a specific type of logarithm with the base of the mathematical constant \( e \), approximately equal to 2.71828. The natural logarithm is especially useful in calculus and scientific calculations because \( e \) is an irrational number with special mathematical properties.
One of the key characteristics of the natural logarithm is that it simplifies the differentiation and integration processes.

• For example, the derivative of \( \ln x \) is \( \frac{1}{x} \).
• On the other hand, the integral of \( \frac{1}{x} \) is \( \ln |x| + C \), where \( C \) is the constant of integration.

In practical use, natural logarithms often appear in growth models, decay processes, and continuous compounding in finance. Recognizing \( \ln \) in mathematical problems can streamline solving them.

Calculator Approximation

Using a calculator to approximate logarithmic values is crucial for getting quick answers, especially when working with natural logarithms. Most scientific calculators have a specific button for \( \ln \), allowing us to easily calculate the logarithms of various numbers.
When dealing with expressions like \( \ln 12 + \ln 5 \) and \( \ln 60 \), following these steps can be helpful:

• First, find \( \ln 12 \) and \( \ln 5 \) separately using the calculator. You'll get approximate values of 2.4849 and 1.6094, respectively.
• Then, add these values to get \( \ln 12 + \ln 5 \approx 4.0943 \).
• Calculate \( \ln 60 \) separately, which also turns out to be approximately 4.0943.

Having this capability is valuable for checking the accuracy of manual calculations or when learning to apply logarithmic properties.

Mathematical Expressions

Mathematical expressions are a combination of numbers, variables, and mathematical operators that together form a quantity or relationship. In the context of logarithms, mathematical expressions often involve calculating powers and roots, which can be complex without the correct methods.
Logarithmic expressions can look intimidating but understanding properties like the product property simplifies them significantly. Consider expressions involving \( \ln \), for instance:

• \( \ln(a) + \ln(b) \) transforms into \( \ln(ab) \).
• Such transformations help in reducing the complexity of problems.

Aside from simplification, understanding mathematical expressions involves recognizing their components and knowing how they can be rewritten for easier calculation. Logarithms, particularly the \( \ln \), play an essential role in expressing relationships in exponential growth and decay models.