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Answer :
To find an expression equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex], we can simplify it by understanding logarithms.
1. Evaluate [tex]\(\log_2 128\)[/tex]:
- [tex]\(128\)[/tex] can be expressed as a power of [tex]\(2\)[/tex]: [tex]\(128 = 2^7\)[/tex].
- Therefore, [tex]\(\log_2 128 = 7\)[/tex].
2. Evaluate [tex]\(\log_2 16\)[/tex]:
- [tex]\(16\)[/tex] can also be expressed as a power of [tex]\(2\)[/tex]: [tex]\(16 = 2^4\)[/tex].
- Hence, [tex]\(\log_2 16 = 4\)[/tex].
3. Compute the given expression:
- Using the evaluations above, [tex]\(\frac{\log_2 128}{\log_2 16} = \frac{7}{4}\)[/tex].
We know from the properties of logarithms that:
[tex]\(\log_b a = \frac{\log_c a}{\log_c b}\)[/tex].
In this case:
- The expression [tex]\(\frac{7}{4}\)[/tex] is equal to [tex]\(\log_{16} 128\)[/tex].
Thus, the expression equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex]. Therefore, the correct answer is:
[tex]\(\log_{16} 128\)[/tex].
1. Evaluate [tex]\(\log_2 128\)[/tex]:
- [tex]\(128\)[/tex] can be expressed as a power of [tex]\(2\)[/tex]: [tex]\(128 = 2^7\)[/tex].
- Therefore, [tex]\(\log_2 128 = 7\)[/tex].
2. Evaluate [tex]\(\log_2 16\)[/tex]:
- [tex]\(16\)[/tex] can also be expressed as a power of [tex]\(2\)[/tex]: [tex]\(16 = 2^4\)[/tex].
- Hence, [tex]\(\log_2 16 = 4\)[/tex].
3. Compute the given expression:
- Using the evaluations above, [tex]\(\frac{\log_2 128}{\log_2 16} = \frac{7}{4}\)[/tex].
We know from the properties of logarithms that:
[tex]\(\log_b a = \frac{\log_c a}{\log_c b}\)[/tex].
In this case:
- The expression [tex]\(\frac{7}{4}\)[/tex] is equal to [tex]\(\log_{16} 128\)[/tex].
Thus, the expression equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex]. Therefore, the correct answer is:
[tex]\(\log_{16} 128\)[/tex].
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