Problem 56 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

\( \ln (47 \times 93) \approx 8.3827 \).

Step 1: Apply Properties of Logarithms

We can simplify this logarithmic expression using the product property of logarithms: \( \ln (A \times B) = \ln A + \ln B \). Therefore, \( \ln (47 \times 93) \) becomes \( \ln 47 + \ln 93 \).

Step 2: Calculate Each Natural Logarithm

Use a calculator to find decimal approximations for each natural logarithm:- \( \ln 47 \approx 3.8501 \)- \( \ln 93 \approx 4.5326 \)

Step 3: Add the Results

Add the two natural logarithm values obtained from the calculator: \( 3.8501 + 4.5326 = 8.3827 \).

Step 4: Present Final Approximation

Therefore, \( \ln (47 \times 93) \approx 8.3827 \).

Key Concepts

Understanding the Properties of Logarithms

Logarithms are incredibly useful in mathematics for transforming multiplicative relationships into additive ones. This is thanks to the properties of logarithms. One of the key properties is the **product property**. This property states that the logarithm of a product is the sum of the logarithms of the individual factors. In formula terms, this is written as:

• \( \ln (A \times B) = \ln A + \ln B \)

This property takes advantage of the fundamental arithmetic operation of addition, making complex calculations more manageable. Using this approach, expressions like \( \ln (47 \times 93) \) can be rewritten and simplified to \( \ln 47 + \ln 93 \), transforming the problem into a simpler addition task. This method is particularly useful when dealing with large numbers or when a calculator is required. By employing properties like these, logarithms simplify many mathematical tasks, rendering them more accessible and less cumbersome. Keep this tool in mind as you tackle similar mathematical challenges.

Exploring the Concept of Decimal Approximation

In mathematics, especially when dealing with irrational numbers or complex calculations, decimal approximations become indispensable. Logarithmic expressions often yield such numbers. For instance, natural logarithms of numbers such as 47 and 93 do not result in neat, whole numbers. Instead, they produce irrational decimals that stretch indefinitely. Consequently, calculators are employed to find these approximations.

• The approximate value of \( \ln 47 \) is around 3.8501
• The approximate value of \( \ln 93 \) is around 4.5326

These decimals offer us a practical way to express logarithmic results without dealing with infinite calculations. Remember that the notion of approximation suggests we are getting very close to the true value of the natural logarithm, although not precisely equating it. For most practical solutions and engineering problems, such decimal approximations are sufficient and widely accepted.

The Product Property in Use

The product property of logarithms is not just an abstract rule but a powerful tool that simplifies complex multiplication into manageable steps. When you have a logarithmic expression involving a product, you can split it into smaller parts thanks to this property. Let's look at the example of \( \ln (47 \times 93) \):

• Using the product property, we rewrite it as \( \ln 47 + \ln 93 \).
• This turns a multiplication inside the logarithm into straightforward addition outside of it.

Ultimately, what you're doing is breaking down a potentially hard-to-calculate problem into simpler pieces. By finding each natural logarithm separately and then adding the results, you take advantage of how logarithms transform multiplicative relationships. This approach not just makes calculations easier but also enhances your understanding of how numbers interact with each other in different mathematical contexts. Understanding and applying the product property allows for more effective problem-solving and offers a glimpse into the beautiful symmetry of mathematics.