Question

Use the Change of Base formula to evaluate logv6 42.

A. Log 6/log 42 ≈ 2.086
B. Log 42/log 6 ≈ 2.086
C. Log 42/log 6 ≈ 0.479
D. Log 6/log 42 ≈ 0.479

Answer 1

To evaluate \( \log_{6} 42 \) using the Change of Base formula, we can express it as:

\[
\log_{6} 42 = \frac{\log_{10} 42}{\log_{10} 6}
\]

or using natural logarithms:

\[
\log_{6} 42 = \frac{\ln 42}{\ln 6}
\]

In both cases, the result will give us the same value, but we'll use the first form for simplicity.

Now, let's determine the possible options:

- **Option A**: \( \log 6 / \log 42 \)
- **Option B**: \( \log 42 / \log 6 \)
- **Option C**: \( \log 42 / \log 6 \)
- **Option D**: \( \log 6 / \log 42 \)

From these options, we can see that \( \log_{6} 42 \) corresponds to \( \frac{\log_{10} 42}{\log_{10} 6} \), which is equivalent to **Option B and Option C**.

To find the numerical value of \( \log_{6} 42 \), let's use approximate values of logarithms:

- \( \log_{10} 42 \approx 1.623 \)
- \( \log_{10} 6 \approx 0.778 \)

Now substituting these into the formula:

\[
\log_{6} 42 \approx \frac{1.623}{0.778} \approx 2.086
\]

Thus, the closest matching option is:

**B. \( \log 42/\log 6 \approx 2.086 \)**.