Problem 26 $$\text { Find each value. If ap... [FREE SOLUTION] (2024)

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Logarithms are incredibly useful in solving complex mathematical problems. One important property of logarithms is the subtraction rule, which states that the difference of two logarithms is the logarithm of the quotient of their arguments. Mathematically, it's written as:
\( \text{log} \text{ } a - \text{log} \text{ } b = \text{log} \text{ } \frac{a}{b} \).
This property allows us to simplify expressions such as \( \text{log} \text{ } 643 - \text{log} \text{ } 287 \) into a single logarithm: \( \text{log} \text{ } \frac{643}{287} \). By understanding and applying this property, you can make lengthy calculations more manageable. Remember:

• When you see subtraction between two logarithms, think of division.
• Simplifying expressions beforehand makes calculation easier.

Simplifying fractions is another critical skill in mathematics, especially when dealing with logarithms. When you simplify the fraction \( \frac{643}{287} \), it results in approximately 2.2390. You achieve this by performing the division, ensuring that both the numerator and the denominator are divided to yield a more straightforward number. Here are some tips:

• First, check if the fraction can be simplified manually. For example, see if the numerator and the denominator have common factors.
• If manual simplification is challenging, use a calculator to get a decimal approximation.
• Simplifying the fraction helps reduce the complexity of the final logarithmic calculation.

As evident from the exercise, understanding fraction simplification is a foundational skill that leads you closer to finding the solution accurately.

Calculators are invaluable tools for performing quick and precise calculations, particularly for complex mathematical problems. In this exercise, the final step involves calculating the logarithm of a number. For instance, after simplifying the fraction to 2.2390, you use the calculator to find \( \text{log} \text{ } 2.2390 \), which results in approximately 0.3504. Here’s how to maximize your calculator use:

• Ensure your calculator is set to the correct mode (common logarithm, base 10, in this case).
• Enter the number accurately to avoid mistakes. Double-check the inputs before pressing the calculate button.
• Understand the functions of your calculator. Features like memory storage can help in large calculations.

Following these steps ensures accurate and timely results, allowing you to focus on understanding the concepts rather than getting bogged down by manual calculations.