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Answer :
To solve the problem of finding which expression is equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex], we need to evaluate each option provided.
### Step 1: Simplify the Original Expression
First, let's evaluate [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex].
- Calculate [tex]\(\log_2 128\)[/tex]:
- Since [tex]\(128 = 2^7\)[/tex], [tex]\(\log_2 128 = 7\)[/tex].
- Calculate [tex]\(\log_2 16\)[/tex]:
- Since [tex]\(16 = 2^4\)[/tex], [tex]\(\log_2 16 = 4\)[/tex].
- Divide [tex]\(\log_2 128\)[/tex] by [tex]\(\log_2 16\)[/tex]:
- [tex]\(\frac{7}{4} = 1.75\)[/tex].
So, [tex]\(\frac{\log_2 128}{\log_2 16} = 1.75\)[/tex].
### Step 2: Evaluate Each Option
Now, let's find the logarithmic values for each given option and see which one equals 1.75:
1. [tex]\(\log_{16} 128\)[/tex]:
- Change of base formula gives us [tex]\(\log_{16} 128 = \frac{\log_2 128}{\log_2 16}\)[/tex].
- As calculated earlier, this is [tex]\(1.75\)[/tex].
2. [tex]\(\log_2 128\)[/tex]:
- We calculated this before as [tex]\(7\)[/tex].
3. [tex]\(\log_4 128\)[/tex]:
- Convert it using change of base: [tex]\(\log_4 128 = \frac{\log_2 128}{\log_2 4}\)[/tex].
- [tex]\(\log_2 4 = 2\)[/tex], so [tex]\(\log_4 128 = \frac{7}{2} = 3.5\)[/tex].
4. [tex]\(\log_{128} 16\)[/tex]:
- Use change of base: [tex]\(\log_{128} 16 = \frac{\log_2 16}{\log_2 128}\)[/tex].
- This is actually [tex]\(\frac{4}{7} \approx 0.571\)[/tex].
### Conclusion
The expression [tex]\(\log_{16} 128\)[/tex] equals 1.75, which matches [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex]. Therefore, the equivalent expression to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex].
### Step 1: Simplify the Original Expression
First, let's evaluate [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex].
- Calculate [tex]\(\log_2 128\)[/tex]:
- Since [tex]\(128 = 2^7\)[/tex], [tex]\(\log_2 128 = 7\)[/tex].
- Calculate [tex]\(\log_2 16\)[/tex]:
- Since [tex]\(16 = 2^4\)[/tex], [tex]\(\log_2 16 = 4\)[/tex].
- Divide [tex]\(\log_2 128\)[/tex] by [tex]\(\log_2 16\)[/tex]:
- [tex]\(\frac{7}{4} = 1.75\)[/tex].
So, [tex]\(\frac{\log_2 128}{\log_2 16} = 1.75\)[/tex].
### Step 2: Evaluate Each Option
Now, let's find the logarithmic values for each given option and see which one equals 1.75:
1. [tex]\(\log_{16} 128\)[/tex]:
- Change of base formula gives us [tex]\(\log_{16} 128 = \frac{\log_2 128}{\log_2 16}\)[/tex].
- As calculated earlier, this is [tex]\(1.75\)[/tex].
2. [tex]\(\log_2 128\)[/tex]:
- We calculated this before as [tex]\(7\)[/tex].
3. [tex]\(\log_4 128\)[/tex]:
- Convert it using change of base: [tex]\(\log_4 128 = \frac{\log_2 128}{\log_2 4}\)[/tex].
- [tex]\(\log_2 4 = 2\)[/tex], so [tex]\(\log_4 128 = \frac{7}{2} = 3.5\)[/tex].
4. [tex]\(\log_{128} 16\)[/tex]:
- Use change of base: [tex]\(\log_{128} 16 = \frac{\log_2 16}{\log_2 128}\)[/tex].
- This is actually [tex]\(\frac{4}{7} \approx 0.571\)[/tex].
### Conclusion
The expression [tex]\(\log_{16} 128\)[/tex] equals 1.75, which matches [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex]. Therefore, the equivalent expression to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex].
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