Question

Zahra is finding an approximation to four decimal places of log11. What is the upper bound approximation?
A. 1.04140
B. 1.041393
C. 1.0413
D. 1.04139

Answer 1

To find the upper bound approximation for \( \log_{11} \), we can use a calculator or logarithm properties. However, since we need a specific four decimal place approximation, we can estimate it based on common logarithm tables or calculators.

The value of \( \log_{11} \) can be calculated using the change of base formula:

\[
\log_{11} = \frac{\log_{10}(11)}{\log_{10}(10)}
\]

Using a calculator:

\[
\log_{10}(11) \approx 1.041392685
\]

Thus,

\[
\log_{11} \approx 1.041392685
\]

Rounding to four decimal places, this gives:

\[
\log_{11} \approx 1.0414
\]

For the options provided, we compare them to find the upper bound:

A. 1.04140 (this matches \( 1.0414 \))
B. 1.041393 (this is greater than \( 1.041392685 \))
C. 1.0413 (this is below \( 1.041392685 \))
D. 1.04139 (this is also below \( 1.041392685 \))

In the context of upper bounds, the correct choice that stays within the value of \( \log_{11} \) and rounds to four decimal places is:

**D. 1.04139**