Problem 56 - A Graphical Approach to College Algebra

Question

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln (47 \times 93)$$

Step-by-Step Solution

Answer

The decimal approximation is 8.379.

Step 1: Simplify the expression inside the logarithm

First, let's simplify the expression inside the logarithm by multiplying the two numbers: 47 and 93.\[ 47 \times 93 = 4371 \]

Step 2: Use the logarithm property to separate the calculation

We know the property of logarithms that states: \( \ln(a \times b) = \ln(a) + \ln(b) \). However, since we have already multiplied the numbers, we can proceed directly to find \( \ln(4371) \).

Step 3: Calculate the natural logarithm using a calculator

Using a calculator, input the value 4371 and compute its natural logarithm to find \( \ln(4371) \).

Step 4: Record the decimal approximation

After calculating, we find that:\[ \ln(4371) \approx 8.379 \]

Key Concepts

Logarithm Properties

Logarithm properties are fundamental rules that simplify complex logarithmic expressions and enable us to solve them more easily. One of the most important properties is the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of its factors. Mathematically, this can be expressed as: \[ \ln(a \times b) = \ln(a) + \ln(b) \]This property is particularly useful when dealing with natural logarithms, such as \( \ln(47 \times 93) \). By using this rule, you could independently calculate the logarithm of each number and then sum the results to find your answer. The product rule helps break down complex problems into simpler parts, aiding in better understanding and computation.

Decimal Approximation

Decimal approximation is the process of finding a close approximation of a number in decimal form, especially when dealing with irrational numbers like natural logarithms. Outputs from logarithms often result in irrational numbers that cannot be exactly expressed as simple fractions or finite decimals; hence a decimal approximation is used. For instance, when you calculate \( \ln(4371) \), the result is approximately \( 8.379 \). This value is rounded to three decimal places for simplicity and practicality, making it easier to use in further calculations or comparisons. Decimal approximations provide an accessible way to work with otherwise complex numbers. It's important to note that while the approximation is close, it is not the exact value, thus should be used carefully in sensitive mathematical operations.

Calculator Usage

Using a calculator when dealing with logarithms is a common practice, especially for complex numbers that are not easily simplified. Modern calculators, including those found on smartphones and computers, have built-in functions to calculate natural logarithms quickly. To find \( \ln(4371) \), you would typically:

• Turn on your calculator and access the logarithm functions, usually marked as 'ln' for natural logarithms.
• Enter the value 4371 and press the 'ln' button.
• Read the result that the calculator displays, which is \( 8.379 \) in this case.

Calculators are essential tools that help eliminate mechanical errors in calculations and save time, particularly when dealing with numbers that are not easily manageable manually.