Question

Which of the following is equivalent to log216
?

Option #1: 3log6

Option #2: log6⋅log6⋅log6

Option #3: 36log6

(1 point)
Option #

Answer 1

To find which of the options is equivalent to \( \log_2 16 \), we can first simplify \( \log_2 16 \).

Since \( 16 = 2^4 \), we can use the logarithmic identity:
\[
\log_b (a^n) = n \cdot \log_b a
\]

Thus, we have:
\[
\log_2 16 = \log_2 (2^4) = 4 \cdot \log_2 2 = 4
\]

Next, we need to compare this value to the given options. To match the expression, we will convert both sides to base 6 if necessary and see if any option equals 4.

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

2. **Option #2: \( \log 6 \cdot \log 6 \cdot \log 6 = (\log 6)^3 \)**

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

To find which of these options equals 4, we will recall that \( \log_6 x \) can be converted using the change of base formula:
\[
\log_6 x = \frac{\log_2 x}{\log_2 6}
\]
Thus, \( \log 6 \) is not a simple value.

Now, let's check each option:

1. **For Option #1:** \( 3 \log 6 \) does not equal 4 because \( \log 6 \) is a positive value but not specific.

2. **For Option #2:** \( (\log 6)^3 \) is even smaller and will not equal 4.

3. **For Option #3:** If \( 36 \log 6 \) equals 4, then we would need:
\[
36 \log 6 = 4 \implies \log 6 = \frac{4}{36} = \frac{1}{9}
\]
This would need to be checked but we can see that it is not a simple result either.

Without specific calculations for \( \log_6 \) and substituting exact values, we find that none directly equal 4 based on initial appearances without more information on the logarithm base value of 6.

Since none of the calculations simplify neatly to 4, I'd recommend evaluating \( \log 6 \) estimates based on known values or other methods, but typically, logarithmic multiples aren't transforming nicely in this context.

Thus, the best confirmations yield:
Since there is no exact alternative provided directly leading to 4 through the base interpretations suggested and numerically unless you get root estimates, thus logically, **none of the options are simplifying to \( 4 \).**

Nonetheless, in a multiple-choice setup, typically Context or additional resource references help clarify specific logarithmic conversions denoting options equivalently.

So, theoretically but more accurately none truly match, or an error in choice context on the estimations seems likely. Please verify or provide nature/context based more for detailed validation.

Overall answer/recommendation: "None of the options are correct if modeling pure logarithmic transformation accurately."