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Answer :
To solve the problem, we need to identify which of the expressions is equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex].
Let's follow these steps:
1. Understand the expression: We have [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex]. This expression can be interpreted using the change of base formula or by directly calculating the logarithmic values.
2. Calculate individual logarithms:
- [tex]\(\log_2 128\)[/tex]: Since [tex]\(128 = 2^7\)[/tex], we have [tex]\(\log_2 128 = 7\)[/tex].
- [tex]\(\log_2 16\)[/tex]: Since [tex]\(16 = 2^4\)[/tex], we have [tex]\(\log_2 16 = 4\)[/tex].
3. Compute the division: Calculate [tex]\(\frac{\log_2 128}{\log_2 16} = \frac{7}{4} = 1.75\)[/tex].
4. Match with given options: The result [tex]\(1.75\)[/tex] corresponds to a logarithmic expression equal to [tex]\(1.75\)[/tex]. We need to find out which expression from the options has a value of [tex]\(1.75\)[/tex].
Now, let's evaluate each option:
- Option 1: [tex]\(\log_{128} 16\)[/tex] would be equivalent to [tex]\(\frac{\log 16}{\log 128}\)[/tex].
- Option 2: [tex]\(\log_{16} 128\)[/tex] would be equivalent to [tex]\(\frac{\log 128}{\log 16}\)[/tex].
- Option 3: [tex]\(\log_4 128\)[/tex] does not match [tex]\(\frac{7}{4}\)[/tex].
- Option 4: [tex]\(\log_2 128\)[/tex] equals [tex]\(7\)[/tex], and clearly, it does not match [tex]\(1.75\)[/tex].
From these evaluations, [tex]\(\log_{16} 128\)[/tex] simplifies to [tex]\(\frac{\log 128}{\log 16}\)[/tex], which matches our division result of [tex]\(\frac{\log_2 128}{\log_2 16} = 1.75\)[/tex].
Therefore, the correct expression equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex].
Let's follow these steps:
1. Understand the expression: We have [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex]. This expression can be interpreted using the change of base formula or by directly calculating the logarithmic values.
2. Calculate individual logarithms:
- [tex]\(\log_2 128\)[/tex]: Since [tex]\(128 = 2^7\)[/tex], we have [tex]\(\log_2 128 = 7\)[/tex].
- [tex]\(\log_2 16\)[/tex]: Since [tex]\(16 = 2^4\)[/tex], we have [tex]\(\log_2 16 = 4\)[/tex].
3. Compute the division: Calculate [tex]\(\frac{\log_2 128}{\log_2 16} = \frac{7}{4} = 1.75\)[/tex].
4. Match with given options: The result [tex]\(1.75\)[/tex] corresponds to a logarithmic expression equal to [tex]\(1.75\)[/tex]. We need to find out which expression from the options has a value of [tex]\(1.75\)[/tex].
Now, let's evaluate each option:
- Option 1: [tex]\(\log_{128} 16\)[/tex] would be equivalent to [tex]\(\frac{\log 16}{\log 128}\)[/tex].
- Option 2: [tex]\(\log_{16} 128\)[/tex] would be equivalent to [tex]\(\frac{\log 128}{\log 16}\)[/tex].
- Option 3: [tex]\(\log_4 128\)[/tex] does not match [tex]\(\frac{7}{4}\)[/tex].
- Option 4: [tex]\(\log_2 128\)[/tex] equals [tex]\(7\)[/tex], and clearly, it does not match [tex]\(1.75\)[/tex].
From these evaluations, [tex]\(\log_{16} 128\)[/tex] simplifies to [tex]\(\frac{\log 128}{\log 16}\)[/tex], which matches our division result of [tex]\(\frac{\log_2 128}{\log_2 16} = 1.75\)[/tex].
Therefore, the correct expression equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex].
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