Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Approximate the logarithm using the properties of logarithms, given \(\log _{b} 2 \approx 0.3562\), \(\log _{b} 3 \approx 0.5646\), and \(\log _{b} 5 \approx 0.8271\)

Step-by-Step Solution

Answer

The approximate value of \( \log _{b} 30 \) is 1.7479.

Step 1: Breakdown of 30

30 can be broken down into the product of the numbers 2, 3, and 5 for which the logarithms have been given. Therefore, \( \log _{b} 30 = \log _{b} (2 \cdot 3 \cdot 5) = \log _{b} 2 + \log _{b} 3 + \log _{b} 5 \).

Step 2: Substitute Values

The values of \( \log _{b} 2 \), \( \log _{b} 3 \), and \( \log _{b} 5 \) are given. We can substitute these values into the equation from Step 1. So \( \log _{b} 30 = 0.3562 + 0.5646 + 0.8271 \).

Step 3: Evaluate sum

The sum 0.3562 + 0.5646 + 0.8271 is 1.7479. Therefore, \( \log _{b} 30 \approx 1.7479 \).

Key Concepts

Understanding Logarithmic Identities

Logarithmic identities are fundamental tools that help simplify and solve logarithmic expressions with ease. One of the most common identities is the

• Product Rule: This states that the logarithm of a product is the sum of the logarithms of its factors. In formula terms, it's expressed as: \[\log_b (MN) = \log_b M + \log_b N\]This identity was used in the original exercise to break down \( \log_b 30 \) into simpler parts.
• Quotient Rule: This rule is used when dealing with the division of numbers. It says the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator:\[\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\]
• Power Rule: If you have an exponent, this rule is necessary. It simplifies the logarithm by bringing the exponent in front:\[\log_b (M^n) = n \cdot \log_b M\]

These rules are crucial for managing complex logarithmic equations and make calculations manageable and structured.

Logarithm Calculation Essentials

When tasked with calculating logarithms without a calculator, understanding how to apply logarithmic identities effectively is key. For example, in the original exercise, the given values for \(\log_b 2\), \(\log_b 3\), and \(\log_b 5\) were used to calculate \(\log_b 30\) by recognizing that 30 is a product of these numbers.
This allowed us to apply the product rule directly. Let's break down this calculation:

• First, notice that 30 can be factored as \(2 \times 3 \times 5\).
• Apply the product rule: \(\log_b (2 \times 3 \times 5) = \log_b 2 + \log_b 3 + \log_b 5\).
• Then substitute each logarithm with the given approximations.
• Add them together to find that \(\log_b 30 \approx 1.7479\).

Breaking down logarithms into sums or differences using these identities streamlines the calculation process.

Approximation Methods for Logarithms

In many practical scenarios, exact logarithmic values are unknown, necessitating approximation methods. These methods are beneficial when values are not easily calculated by hand.

Using Given Values:

One effective method is using known approximations. Like in our problem, using the given values for smaller components (of \(\log_b 2\), \(\log_b 3\), and \(\log_b 5\)) allows us to construct and approximate larger, more complex logarithmic expressions.

Linear Approximation:

This involves using a tangent line to display function values. While not demonstrated in our example, it's useful for small differences between known values and those needing approximation.

• It forms the basis for understanding minor changes around a known point and applying a linear equation for approximation.

These methods together provide a robust approach to handling tasks where exact logarithmic values are impractical to derive, ensuring approximations are efficiently handled.