To find which expression is equivalent to [tex]\(\log_2 n = 4\)[/tex], let's solve the equation step-by-step and evaluate the given choices.
1. Solve [tex]\(\log_2 n = 4\)[/tex]:
The equation [tex]\(\log_2 n = 4\)[/tex] means that [tex]\(n\)[/tex] is the number such that [tex]\(\log_2 n\)[/tex] equals 4 in base 2. This can be written as:
[tex]\[ 2^4 = n \][/tex]
Calculating [tex]\(2^4\)[/tex]:
[tex]\[ n = 16 \][/tex]
2. Evaluating the given choices with [tex]\(n = 16\)[/tex]:
We need to find which expression matches [tex]\(\log n = 4 \log 2\)[/tex].
Let's evaluate each choice:
- Choice 1: [tex]\(\log n = \frac{\log 2}{4}\)[/tex]
[tex]\(\log n\)[/tex] is the logarithm of [tex]\(n\)[/tex]. With [tex]\(n = 16\)[/tex]:
[tex]\[ \log 16 \neq \frac{\log 2}{4} \][/tex]
- Choice 2: [tex]\(n = \frac{\log 2}{\log 4}\)[/tex]
This is not an appropriate form for the logarithmic expression we are dealing with. It does not correspond to a logarithmic transformation that results in [tex]\(n = 16\)[/tex].
- Choice 3: [tex]\(n = \log 4 \cdot \log 2\)[/tex]
This implies [tex]\( n \)[/tex] is a product of logarithms, but we know [tex]\( n = 16 \)[/tex]. So:
[tex]\[ 16 \neq \log 4 \cdot \log 2 \][/tex]
- Choice 4: [tex]\(\log n = 4 \log 2\)[/tex]
If we take [tex]\( n = 16 \)[/tex] and substitute it:
[tex]\[ \log 16 = 4 \log 2 \][/tex]
We need to check this:
[tex]\[ \log 16 \text{ in base 10} = \log (2^4) = 4 \log 2 \][/tex]
This is indeed true since the laws of logarithms confirm this transformation.
Thus, the correct and equivalent expression to [tex]\(\log_2 n = 4\)[/tex] is:
[tex]\[ \boxed{\log n = 4 \log 2} \][/tex]
