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Answer :
To solve the problem of identifying which expression is equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex], let's understand what each log expression means and evaluate its value:
1. Evaluate [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex]:
- [tex]\(\log_2 128\)[/tex] gives the power to which the base 2 must be raised to get 128. Since [tex]\(128 = 2^7\)[/tex], [tex]\(\log_2 128 = 7\)[/tex].
- [tex]\(\log_2 16\)[/tex] gives the power to which the base 2 must be raised to get 16. Since [tex]\(16 = 2^4\)[/tex], [tex]\(\log_2 16 = 4\)[/tex].
- Thus, [tex]\(\frac{\log_2 128}{\log_2 16} = \frac{7}{4} = 1.75\)[/tex].
Now let's evaluate the four options:
2. Evaluate [tex]\(\log_4 128\)[/tex]:
- The expression [tex]\(\log_4 128\)[/tex] asks for the power to which 4 must be raised to get 128. After calculation, this value is 3.5.
3. Evaluate [tex]\(\log_{128} 16\)[/tex]:
- The expression [tex]\(\log_{128} 16\)[/tex] asks for the power to which 128 must be raised to get 16. After calculation, this value is approximately 0.5714.
4. Evaluate [tex]\(\log_2 128\)[/tex]:
- We've already calculated [tex]\(\log_2 128 = 7\)[/tex].
5. Evaluate [tex]\(\log_{16} 128\)[/tex]:
- The expression [tex]\(\log_{16} 128\)[/tex] asks for the power to which 16 must be raised to get 128. This value is 1.75, which matches our original expression.
Therefore, the expression equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex].
1. Evaluate [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex]:
- [tex]\(\log_2 128\)[/tex] gives the power to which the base 2 must be raised to get 128. Since [tex]\(128 = 2^7\)[/tex], [tex]\(\log_2 128 = 7\)[/tex].
- [tex]\(\log_2 16\)[/tex] gives the power to which the base 2 must be raised to get 16. Since [tex]\(16 = 2^4\)[/tex], [tex]\(\log_2 16 = 4\)[/tex].
- Thus, [tex]\(\frac{\log_2 128}{\log_2 16} = \frac{7}{4} = 1.75\)[/tex].
Now let's evaluate the four options:
2. Evaluate [tex]\(\log_4 128\)[/tex]:
- The expression [tex]\(\log_4 128\)[/tex] asks for the power to which 4 must be raised to get 128. After calculation, this value is 3.5.
3. Evaluate [tex]\(\log_{128} 16\)[/tex]:
- The expression [tex]\(\log_{128} 16\)[/tex] asks for the power to which 128 must be raised to get 16. After calculation, this value is approximately 0.5714.
4. Evaluate [tex]\(\log_2 128\)[/tex]:
- We've already calculated [tex]\(\log_2 128 = 7\)[/tex].
5. Evaluate [tex]\(\log_{16} 128\)[/tex]:
- The expression [tex]\(\log_{16} 128\)[/tex] asks for the power to which 16 must be raised to get 128. This value is 1.75, which matches our original expression.
Therefore, the expression equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex].
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