Question

Which of the following is equivalent to log216
?

Option #1: 3log6


Option #2: log6⋅log6⋅log6


Option #3: 36log6

Answer 1

To determine which of the given options is equivalent to \(\log_2 16\), we first evaluate \(\log_2 16\).

Since \(16\) can be expressed as \(2^4\), we can rewrite the logarithm:

\[
\log_2 16 = \log_2 (2^4) = 4.
\]

Now, let's evaluate each of the options to see which one equals \(4\).

**Option #1: \(3 \log 6\)**

This option does not give a clear numeric value without knowing what \(\log 6\) is. We can't directly compare it to \(4\) unless we calculate it. So let's check this option numerically:

If we use the change of base formula:
\[
\log 6 \approx 0.7781 \quad (\text{using base 10 or the natural logarithm}),
\]
Then,
\[
3 \log 6 \approx 3 \times 0.7781 \approx 2.3343 \quad (\text{which is not } 4).
\]

**Option #2: \(\log 6 \cdot \log 6 \cdot \log 6\)**

This option denotes \( (\log 6)^3 \). Using the approximate value again:
\[
(\log 6)^3 \approx (0.7781)^3 \approx 0.4743 \quad (\text{which is not } 4).
\]

**Option #3: \(36 \log 6\)**

Using the same approximation:
\[
36 \log 6 \approx 36 \times 0.7781 \approx 28.06 \quad (\text{which is not } 4).
\]

After checking all options, none of them evaluated to \(4\).

Therefore, it appears that none of the provided options are equivalent to \(\log_2 16\). If you are looking for correctness based on the logarithmic values given, all evaluate to something other than \(4\). Thus the correct option is that none of the options are equivalent to \(\log_2 16\).