Problem 2 - Algebra 2

Question

Use log \(_{3} 2 \approx 0.6309\) and \(\log _{3} 7 \approx 1.7712\) to approximate the value of each expression. \(\log _{3} 14\)

Step-by-Step Solution

Answer

The approximate value of \( \log_{3} 14 \) is 2.4021.

Step 1: Express the Logarithm in Terms of Known Logs

Given the expression \( \log_{3} 14 \), we recognize that 14 can be written as the product of 2 and 7. Therefore, we can use the logarithmic property \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \). Thus, \( \log_{3} 14 = \log_{3}(2 \times 7) = \log_{3} 2 + \log_{3} 7 \).

Step 2: Substitute Known Values

We substitute the given approximate values into the expression: \( \log_{3} 2 + \log_{3} 7 = 0.6309 + 1.7712 \).

Step 3: Perform the Addition

Add the two approximate values together. \( 0.6309 + 1.7712 = 2.4021 \). This is the approximate value of \( \log_{3} 14 \).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for simplifying and solving various mathematical problems. One of the fundamental properties is the product rule for logarithms, which states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, this is expressed as:

• \( \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \)

This property is useful when you know the logarithms of certain numbers and need to find the logarithm of their product.

Besides the product rule, there are other important properties of logarithms:

• Quotient Rule: \( \log_{b}(x/y) = \log_{b}(x) - \log_{b}(y) \)
• Power Rule: \( \log_{b}(x^{n}) = n \cdot \log_{b}(x) \)
• Change of Base: \( \log_{a}(x) = \frac{\log_{b}(x)}{\log_{b}(a)} \)

By using these properties, you can manipulate and solve logarithmic expressions in a more straightforward manner.

Approximating Logarithmic Values

There are instances where exact logarithmic values are difficult to compute, especially without a calculator. In such cases, approximating logarithmic values using known values and properties becomes helpful. For example, if we know \( \log_{3} 2 \approx 0.6309 \) and \( \log_{3} 7 \approx 1.7712 \), we can use these values to find the approximate logarithm of numbers that can be expressed in terms of 2 and 7.

Using the product property, \( \log_{3} 14 = \log_{3} (2 \times 7) \), we substitute the approximate values into the expression:

• Calculate \( \log_{3} 2 + \log_{3} 7 = 0.6309 + 1.7712 \)

This gives the approximate value of \( \log_{3} 14 \) as 2.4021. Approximations are critically useful when exact values are not required or when they are impossible to derive without computational aid.

Logarithmic Calculations

Logarithmic calculations involve manipulating expressions using logarithm properties to achieve a simplified form or find an unknown value. Often, these calculations are facilitated by approximate values when precise values aren’t known or practical to compute. Let's consider the example of calculating \( \log_{3} 14 \):

In the solution, we utilized the known approximate logarithmic values of 2 and 7. By applying the product property, the calculation of \( \log_{3} (2 \times 7) \) transformed into a simple addition problem of two approximations:

• Calculation: \( 0.6309 + 1.7712 = 2.4021 \)

This approach saves time and effort, especially when handling complex logarithmic equations. Understanding these techniques and approaches enable smoother and more efficient problem solving, allowing for the handling of a wide variety of logarithmic problems in different scenarios.