To find which of the given options is equivalent to \( \log_2 16 \), we can solve for \( \log_2 16 \) first.
We know that: \[ 16 = 2^4 \]
Using the properties of logarithms, we can simplify \( \log_2 16 \): \[ \log_2 16 = \log_2 (2^4) = 4 \cdot \log_2 2 = 4 \]
Now we have \( \log_2 16 = 4 \).
Next, let's evaluate each option to see which one also equals 4.
Option #1: \(3 \log_6\)
This option does not provide us with a numeric value without knowing what \( \log_6 \) is equal to. Generally, we cannot simplify this to equal 4 without additional context about the base or value.
Option #2: \( \log_6 \cdot \log_6 \cdot \log_6 \)
This is \( (\log_6)^3 \). Again, this does not directly equal to 4 without further information regarding the base of \( 6 \).
Option #3: \(36 \log_6\)
This option may equal 4 based on the value of \( \log_6 \): \[ 36 \log_6 = 4 \implies \log_6 = \frac{4}{36} = \frac{1}{9} \]
This answer is also not correct without further context.
Consequently, none of the provided options can be definitively stated as equivalent to \( \log_2 16 \) without additional information about \( \log_6 \).
If you are looking for a quick solution, since we have: \[ \log_2 16 = 4 \]
The equivalences provided do not seem to hold under known logarithmic identities without applying further calculations or context, therefore it's crucial to double-check the definitions or intended context of \( \log_6 \).
In conclusion, none of the options are equivalent to \( \log_2 16 \) based on standard log properties and the base identities unless more context or constraints are provided about \( \log_6 \).
1 answer