Problem 50 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The approximate value of \( \log(47 \times 93) \) is 3.6406.

Step 1: Understand the Problem

The problem requires finding the logarithm of the product 47 and 93, specifically using base 10, which is the common logarithm. We will use a calculator to find the decimal approximation.

Step 2: Use Logarithm Property

According to logarithmic properties, \( \log(a \times b) = \log(a) + \log(b) \). In this case, we use \( a = 47 \) and \( b = 93 \). Therefore, \( \log(47 \times 93) = \log(47) + \log(93) \).

Step 3: Calculate \( \log(47) \)

Using a calculator, find \( \log(47) \). Suppose the approximation given by our calculator is \( \log(47) \approx 1.6721 \).

Step 4: Calculate \( \log(93) \)

Using a calculator, find \( \log(93) \). Let's say the approximation is \( \log(93) \approx 1.9685 \).

Step 5: Add the Results

Now, add the logarithms found in steps 3 and 4. So, \( \log(47) + \log(93) \approx 1.6721 + 1.9685 = 3.6406 \).

Step 6: Verify with Direct Calculation

To verify, compute \( 47 \times 93 = 4371 \). Then compute \( \log(4371) \) using a calculator, which should approximately equal 3.6406, confirming our solution.

Key Concepts

Decimal Approximation

When dealing with logarithms, often we need precise values that are not readily available. This is where decimal approximations come into play. A decimal approximation gives us a nearby value to the exact number, which is especially handy for quick calculations or when using logarithms in practical applications.
In the exercise, after applying logarithm properties, we calculate the logarithms of individual numbers and use a calculator to approximate these values. The results are rounded decimal numbers. For instance:

• The approximation of \( \log(47) \) is 1.6721
• The approximation of \( \log(93) \) is 1.9685

These approximations allow us to sum them up conveniently as \( \log(47)+\log(93) \approx 3.6406 \). Decimal approximations are essential in digital computations, helping us to achieve a balance between precision and efficiency.

Logarithmic Properties

Logarithmic properties are powerful tools in simplifying calculations and turning complex problems into manageable tasks. One key property utilized often is the Product Rule for logarithms. This rule states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, it is expressed as:\[ \log(a \times b) = \log(a) + \log(b) \]In our exercise, instead of directly trying to find \(\log(47 \times 93)\), we split it using this property. By calculating \(\log(47)\) and \(\log(93)\) separately, then adding them together, we simplified the computation.
Moreover, these properties not only make difficult calculations easier, but also enhance comprehension of logarithms as mathematical operations, making it easier to handle them in both theoretical and practical contexts.

Common Logarithm

Common logarithms are logarithms with base 10, typically denoted as \( \log \). In many situations, especially in science and engineering, base 10 logarithms are most commonly used, hence the name 'common logarithm.' Unlike natural logarithms that have \( \ln \) with base \(e\), common logarithms form the basis of many logarithmic calculations in everyday life.
In our exercise, we used the common logarithm to compute \(\log(47 \times 93)\). Because calculators default to base 10 for \(\log\), it makes finding these values straightforward, and it utilizes the conceptual framework students learn when first encountering logarithms. Understanding and applying common logarithms is crucial, as it lays the foundation for further learning in mathematics and various related fields.