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Answer :
Sure! Let's go through the process step-by-step to identify which expression is equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex].
The expression [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] can be used to compare several options to see which one results in the same value.
First, let's break down the given expression:
1. Calculate [tex]\(\log_2 128\)[/tex]:
- Since [tex]\(2^7 = 128\)[/tex], we have [tex]\(\log_2 128 = 7\)[/tex].
2. Calculate [tex]\(\log_2 16\)[/tex]:
- Since [tex]\(2^4 = 16\)[/tex], we have [tex]\(\log_2 16 = 4\)[/tex].
3. Evaluate the expression [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex]:
- This becomes [tex]\(\frac{7}{4} = 1.75\)[/tex].
Now, let's evaluate the values of each of the given options to find which one equals 1.75:
- [tex]\(\log_{128} 16\)[/tex]:
- This is asking what power we need to raise 128 to get 16.
- Using the result, this value is approximately [tex]\(0.5714\)[/tex], which is not 1.75.
- [tex]\(\log_4 128\)[/tex]:
- This is asking what power we need to raise 4 to get 128.
- Using the result, this value is approximately [tex]\(3.5\)[/tex], which is not 1.75.
- [tex]\(\log_{16} 128\)[/tex]:
- This is asking what power we need to raise 16 to get 128.
- Using the result, this value is [tex]\(1.75\)[/tex], which matches our expression's value.
- [tex]\(\log_2 128\)[/tex]:
- We already identified this as 7, which does not match 1.75.
Therefore, the expression equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex].
The expression [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] can be used to compare several options to see which one results in the same value.
First, let's break down the given expression:
1. Calculate [tex]\(\log_2 128\)[/tex]:
- Since [tex]\(2^7 = 128\)[/tex], we have [tex]\(\log_2 128 = 7\)[/tex].
2. Calculate [tex]\(\log_2 16\)[/tex]:
- Since [tex]\(2^4 = 16\)[/tex], we have [tex]\(\log_2 16 = 4\)[/tex].
3. Evaluate the expression [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex]:
- This becomes [tex]\(\frac{7}{4} = 1.75\)[/tex].
Now, let's evaluate the values of each of the given options to find which one equals 1.75:
- [tex]\(\log_{128} 16\)[/tex]:
- This is asking what power we need to raise 128 to get 16.
- Using the result, this value is approximately [tex]\(0.5714\)[/tex], which is not 1.75.
- [tex]\(\log_4 128\)[/tex]:
- This is asking what power we need to raise 4 to get 128.
- Using the result, this value is approximately [tex]\(3.5\)[/tex], which is not 1.75.
- [tex]\(\log_{16} 128\)[/tex]:
- This is asking what power we need to raise 16 to get 128.
- Using the result, this value is [tex]\(1.75\)[/tex], which matches our expression's value.
- [tex]\(\log_2 128\)[/tex]:
- We already identified this as 7, which does not match 1.75.
Therefore, the expression equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex].
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