Problem 58 - Precalculus with Calculus Previews

Question

In Problems \(57-68\), use \(\log _{b} 4=0.6021\) and \(\log _{b} 5=\) \(0.6990\) to evaluate the given logarithm. Round your answer to four decimal places. $$ \log _{b} 20 $$

Step-by-Step Solution

Answer

\( \log_b 20 \approx 1.3011 \).

Step 1: Understanding the problem

We need to calculate the value of \( \log_b 20 \) given that \( \log_b 4 = 0.6021 \) and \( \log_b 5 = 0.6990 \).

Step 2: Expressing 20 in terms of 4 and 5

Recognize that 20 can be factored into terms involving 4 and 5: \( 20 = 4 \times 5 \). This means we can express \( \log_b 20 \) as \( \log_b (4 \times 5) \).

Step 3: Using the Product Rule of Logarithms

Apply the product rule which states \( \log_b (MN) = \log_b M + \log_b N \). For our expression, this becomes \( \log_b 20 = \log_b 4 + \log_b 5 \).

Step 4: Substituting the known logarithms

Substitute the given values into the expression: \( \log_b 4 = 0.6021 \) and \( \log_b 5 = 0.6990 \). Therefore, \( \log_b 20 = 0.6021 + 0.6990 \).

Step 5: Calculating the result

Add the two known logarithm values: \( 0.6021 + 0.6990 = 1.3011 \). Therefore, \( \log_b 20 \approx 1.3011 \).

Key Concepts

The Product Rule of Logarithms

The product rule of logarithms is an essential tool for simplifying and solving logarithmic expressions. It states that the logarithm of a product is equal to the sum of the logarithms of its factors. In mathematical terms, for any positive numbers \(M\) and \(N\), and base \(b\), this rule is written as:

• \( \log_b (MN) = \log_b M + \log_b N \)

This property allows us to break down complex logarithmic expressions into simpler, more manageable parts. In practical terms, when you know the logarithms of individual factors, you can easily find the logarithm of their product, as seen in the solution for \( \log_b 20 \). Here, by recognizing that \(20 = 4 \times 5\), we used this rule to express \( \log_b 20 \) as \( \log_b 4 + \log_b 5 \). This provided a straightforward pathway to evaluate the logarithm using known values, making the process efficient and error-free.

Logarithmic Properties

Logarithmic properties are rules that simplify calculations and the manipulation of logarithmic expressions. Some principal properties include:

• Product Rule: \( \log_b (MN) = \log_b M + \log_b N \)
• Quotient Rule: \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \)
• Power Rule: \( \log_b (M^k) = k \cdot \log_b M \)

These properties are powerful tools that enable mathematicians to transform and simplify logarithmic expressions in problem-solving scenarios. In this particular example, the product rule was used because 20 was expressed as a product of 4 and 5, allowing the known logarithmic values to substitute directly into the equation and provide a solution quickly. Understanding these fundamental properties ensures you can tackle a wide variety of logarithmic problems with confidence.

Precalculus Problem-Solving

Precalculus mathematics often involves combining various algebraic concepts and logarithmic principles to solve complex problems. Here, an understanding of factoring numbers, combined with the use of logarithms, made solving \( \log_b 20 \) straightforward.When tackling precalculus problems, remember:

• Break down complex expressions into simpler components using fundamental properties and rules.
• Use known values and substitute them into expressions wherever possible to simplify the solving process.
• Re-check your calculations for arithmetic errors by examining each step carefully.

In this exercise, recognizing \(20 = 4 \times 5\) allowed for the application of the product rule, simplifying the problem into the calculation \( \log_b 4 + \log_b 5 \), producing 1.3011 as a rounded result. Mastering these techniques is crucial for success in advanced mathematics and applications involving logarithms.