Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Given that \(\log 3 \approx 0.4771\) and \(\log 4 \approx\) 0.6021, find the value of each logarithm. $$\log \frac{3}{4}$$

Step-by-Step Solution

Answer

The value of \(\log \frac{3}{4}\) is approximately \(-0.1250\).

Step 1: Use logarithmic property to rewrite given expression

We can rewrite the given logarithm using the logarithmic property as: $$\log \frac{3}{4} = \log 3 - \log 4$$

Step 2: Substitute the given values

Replace the given approximations for \(\log3\) and \(\log4\) in the expression: $$\log \frac{3}{4} = 0.4771 - 0.6021$$

Step 3: Perform the calculation

Subtract the values of \(\log 3\) and \(\log 4\) to find the value of \(\log \frac{3}{4}\): $$\log \frac{3}{4} \approx -0.1250$$ So, the value of \(\log \frac{3}{4}\) is approximately \(-0.1250\).

Key Concepts

Logarithmic Properties

Logarithmic properties are crucial tools in simplifying and solving logarithmic expressions. One essential property is the quotient rule, which states:

• \(\log \frac{a}{b} = \log a - \log b\)

This rule allows us to break down a single logarithm into the subtraction of two separate logarithms. This property is particularly helpful when you have known logarithmic values for the individual numbers, such as in the original exercise with \(\log 3\) and \(\log 4\).

Such properties are derived from the basic definition of logarithms and exponent rules. Understanding these properties enables solving more complex logarithmic equations by reducing them to simpler forms. Other important properties include the product rule (\(\log(ab) = \log a + \log b\)) and the power rule (\(\log a^n = n \log a\)). Each serves a unique purpose, but together, they form the foundation for manipulating and understanding logarithmic expressions.

Subtraction of Logarithms

Subtraction of logarithms is directly linked to the quotient property of logarithms. When you have an expression like \(\log \frac{3}{4}\), the subtraction rule allows you to express it as a difference: \(\log 3 - \log 4\).

This form is beneficial because it breaks down complex expressions into simpler parts. By knowing the approximate values of \(\log 3\) (0.4771) and \(\log 4\) (0.6021), you can straightforwardly compute:

• \(\log \frac{3}{4} = 0.4771 - 0.6021\)
• \(\log \frac{3}{4} \approx -0.1250\)

This subtraction process not only aids in simplification but also enhances the understanding of how logarithms work in dividing numbers. It's a clear application of logarithmic properties to obtain results purely through algebraic manipulation.

By practicing problems involving this property, you can develop a better intuition for recognizing when and how to utilize the subtraction of logarithms effectively.

Approximation in Logarithms

Approximation in logarithms is a significant aspect of logarithmic calculations, especially when exact values are unknown or difficult to determine manually. In many cases, logarithm tables or calculators provide approximate values for common logarithms, such as \(\log 3 \approx 0.4771\) and \(\log 4 \approx 0.6021\).

Approximation is particularly useful in education and real-world applications where precision can be balanced with ease and speed of calculation. When using approximations, it's important to recognize that these values are close estimates and the final results may not be perfectly exact. However, they are generally sufficient for most practical purposes.

For calculations where precision is paramount, more accurate methods or tools might be necessary. Nonetheless, understanding approximations helps solidify a practical approach to logarithmic problems, enabling efficient and effective solutions even without advanced computational resources. As seen in the original exercise, using approximations led to the final result of \(\log \frac{3}{4} \approx -0.1250\), demonstrating their utility in simplifying log problems.