Problem 44 - 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$$\log _{b} \sqrt{5}\)

Step-by-Step Solution

Answer

The approximate value of log_b(sqrt(5)) is 0.41355.

Step 1: Remember Logarithm Properties

Keep in mind the properties of logarithms as they would be crucial in solving this problem. The specific property of interest is - log_a(n^m) = m * log_a(n); this property suggests that the logarithm of n raised to a power m equals m times the logarithm of n.

Step 2: Interpret the Problem

In this problem we are interested in finding the value of log_b(sqrt(5)). We recognize that sqrt(5) is just 5 raised to the power of 1/2. By applying the property of logarithms mentioned earlier, we have log_b(sqrt(5)) equals (1/2)*log_b(5).

Step 3: Apply Value Given

Substitute the given value of log_b(5) which is approximately 0.8271 into the equation. This yields log_b(sqrt(5)) equals (1/2)*0.8271.

Step 4: Compute the Approximation

Calculating (1/2)*0.8271, we obtain approximately 0.41355. Therefore, log_b(sqrt(5)) is equal to 0.41355.

Key Concepts

Understanding the Properties of Logarithms

The properties of logarithms are essential tools in simplifying and solving logarithmic expressions. One of the most fundamental properties is that the logarithm of a power can be rewritten as a product. This property can be expressed as: \( \log_a(n^m) = m \cdot \log_a(n) \). Simply put, it allows us to take an exponent in the argument and bring it in front of the logarithm as a multiplier. This transformation is extremely helpful when dealing with square roots, as seen in this exercise, where \(\sqrt{5}\) is rewritten as \(5^{1/2}\). By using the property, we can reframe \(\log_b (\sqrt{5})\) to \((1/2) \cdot \log_b(5)\), making it simpler to compute using the known logarithm of 5.

Approximation of Logarithms Made Simple

Approximation of logarithms is crucial when dealing with non-integer bases or when precise values are not available. In many cases, you will have to rely on approximate values of certain logarithms, as given for this exercise: \( \log_b 2 \approx 0.3562 \), \( \log_b 3 \approx 0.5646 \), and \( \log_b 5 \approx 0.8271 \). To approximate \( \log_b(\sqrt{5}) \), we apply the property discussed, which allows us to multiply the approximated logarithm of 5 by \(1/2\). This simplification is necessary because the base \(b\) isn't defined, necessitating approximations of calculations. Estimation is thus a valuable skill when working with logarithmic expressions in real-world problems.

Effective Mathematical Problem Solving

Mathematical problem solving often involves strategic use of properties and approximations. It's important to approach each problem methodically, interpreting the question and identifying which properties or approximations to apply. In this exercise, we begin by recognizing that \( \log_b(\sqrt{5}) \) can be simplified using the known properties of logarithms. Breaking down the problem into smaller steps, such as rewriting \( \sqrt{5} \) as \(5^{1/2}\) and then applying the logarithmic property, ensures that the solution is both accessible and correct. Remember to substitute values accurately and conduct the computations step-by-step. This approach to problem solving not only simplifies complex logarithmic problems but also enhances your confidence and proficiency in dealing with logarithms.