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Answer :
Let's solve the given expression step by step:
The expression we need to evaluate is [tex]\(\frac{\log _2 128}{\log _2 16}\)[/tex].
1. Calculate [tex]\(\log_2 128\)[/tex]:
[tex]\(\log_2 128\)[/tex] means we need to find the power to which 2 must be raised to get 128. Since [tex]\(2^7 = 128\)[/tex], we have [tex]\(\log_2 128 = 7\)[/tex].
2. Calculate [tex]\(\log_2 16\)[/tex]:
Similarly, [tex]\(\log_2 16\)[/tex] is the power to which 2 must be raised to get 16. Since [tex]\(2^4 = 16\)[/tex], we find [tex]\(\log_2 16 = 4\)[/tex].
3. Evaluate the expression:
Now, divide the results from the above calculations:
[tex]\[
\frac{\log_2 128}{\log_2 16} = \frac{7}{4} = 1.75
\][/tex]
4. Identify the Equivalent Expression:
According to the properties of logarithms, the expression [tex]\(\frac{\log_b A}{\log_b B}\)[/tex] can be simplified using the change of base formula to [tex]\(\log_B A\)[/tex].
So, [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] simplifies to [tex]\(\log_{16} 128\)[/tex].
Therefore, the expression equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex].
The expression we need to evaluate is [tex]\(\frac{\log _2 128}{\log _2 16}\)[/tex].
1. Calculate [tex]\(\log_2 128\)[/tex]:
[tex]\(\log_2 128\)[/tex] means we need to find the power to which 2 must be raised to get 128. Since [tex]\(2^7 = 128\)[/tex], we have [tex]\(\log_2 128 = 7\)[/tex].
2. Calculate [tex]\(\log_2 16\)[/tex]:
Similarly, [tex]\(\log_2 16\)[/tex] is the power to which 2 must be raised to get 16. Since [tex]\(2^4 = 16\)[/tex], we find [tex]\(\log_2 16 = 4\)[/tex].
3. Evaluate the expression:
Now, divide the results from the above calculations:
[tex]\[
\frac{\log_2 128}{\log_2 16} = \frac{7}{4} = 1.75
\][/tex]
4. Identify the Equivalent Expression:
According to the properties of logarithms, the expression [tex]\(\frac{\log_b A}{\log_b B}\)[/tex] can be simplified using the change of base formula to [tex]\(\log_B A\)[/tex].
So, [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] simplifies to [tex]\(\log_{16} 128\)[/tex].
Therefore, the expression equivalent to [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] is [tex]\(\log_{16} 128\)[/tex].
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