Problem 63 - Algebra 2

Question

Use \(\log _{7} 2 \approx 0.3562\) and \(\log _{7} 3 \approx 0.5646\) to approximate the value of each expression. $$ \log _{7} 16 $$

Step-by-Step Solution

Answer

\( \log_{7} 16 \approx 1.4248 \)

Step 1: Express 16 in Terms of Known Values

First, recognize that the number 16 can be written as a power of 2. Specifically, \( 16 = 2^4 \). This is because multiplying 2 by itself four times gives 16.

Step 2: Apply the Logarithm Power Rule

The logarithm power rule states that \( \log_b (x^n) = n \cdot \log_b (x) \). Apply this rule to express \( \log_{7} 16 \) as \( \log_{7} (2^4) = 4 \cdot \log_{7} 2 \).

Step 3: Substitute the Value of \(\log_{7} 2\)

We know from the problem statement that \( \log_{7} 2 \approx 0.3562 \). Substitute this value into the expression: \( 4 \cdot \log_{7} 2 = 4 \cdot 0.3562 \).

Step 4: Perform the Multiplication

Calculate \( 4 \times 0.3562 \). This multiplies to give approximately 1.4248.

Step 5: Conclusion

Thus, \( \log_{7} 16 \approx 1.4248 \) using the given approximations for \( \log_{7} 2 \).

Key Concepts

Logarithmic Approximation

Logarithmic approximation is a technique for estimating the value of a logarithm when exact calculation is not possible or practical. This often involves using known values to approximate unknowns.
Imagine you have a logarithm like \( \log_{7} 16 \), and the exact value is difficult to determine directly. Here, you use approximations, such as \( \log_{7} 2 \approx 0.3562 \), to derive a close estimate.
Two ways to use approximation include:

• Breaking down numbers into known base components, like expressing 16 as \( 2^4 \).
• Utilizing manipulative mathematical rules, including conversion formulas and computational approximations.

This approach makes logarithmic calculations quicker and can be very useful when dealing with larger or more complex figures. By approximating, you transform a potentially cumbersome problem into a simpler one that's much more manageable.

Power Rule in Logarithms

The power rule in logarithms is a helpful tool that simplifies calculations involving powers. It's expressed as \( \log_b (x^n) = n \cdot \log_b (x) \). This equation tells you that the logarithm of a number raised to a power is equal to the exponent times the logarithm of the base number itself.
By using the power rule, you can transform potentially complex problems into straightforward multiplications.

• For example, with \( \log_{7} 16 \), you recognize 16 as \( 2^4 \), and apply the power rule: \( \log_{7} (2^4) = 4 \cdot \log_{7} 2 \).
• This simplifies the calculation to just multiplying the known \( \log_{7} 2 \) by 4, provided by the problem statement.

This technique highlights the elegance and efficiency possible in mathematical problem solving. The power rule reduces the cognitive load and streamlines the process when dealing with exponential expressions in logarithmic terms.

Logarithmic Base Conversion

Logarithmic base conversion is a method of changing the base of a logarithmic expression so that calculations are more manageable. It's rooted in the formula: \( \log_b (x) = \frac{\log_k (x)}{\log_k (b)} \), where \(k\) is the new base you're converting to.
Base conversion can be especially useful when you either have more knowledge or comfort in calculating with a different base.

• If you're given logarithms like \( \log_{7} 2 \) and find it easier to work in base 10, you could employ this conversion to switch bases for easier computation.
• This is advantageous when solving problems that require interoperability among systems using different bases.

However, in some exercises like the present one, it might be simpler to work within the given base and directly apply meaningful rules. Base conversion remains a valuable tool, expanding the possibilities for problem-solving tactics in logarithmic contexts.