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Answer :
To solve the expression [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex], we can use properties of logarithms to simplify it. Let's go through it step-by-step:
1. Understand What the Expression Means:
The expression [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] represents the ratio between [tex]\(\log_2 128\)[/tex] and [tex]\(\log_2 16\)[/tex].
2. Convert Logarithms Using Change of Base:
One important property of logarithms is that if [tex]\(\log_a b = x\)[/tex], then [tex]\(a^x = b\)[/tex].
3. Evaluate [tex]\(\log_2 128\)[/tex] and [tex]\(\log_2 16\)[/tex]:
- First, calculate [tex]\(\log_2 128\)[/tex].
- Since [tex]\(2^7 = 128\)[/tex], it follows that [tex]\(\log_2 128 = 7\)[/tex].
- Now, calculate [tex]\(\log_2 16\)[/tex].
- Since [tex]\(2^4 = 16\)[/tex], it follows that [tex]\(\log_2 16 = 4\)[/tex].
4. Simplify the Fraction:
- Now, substitute the values into the expression:
[tex]\[
\frac{\log_2 128}{\log_2 16} = \frac{7}{4}
\][/tex]
5. Identify the Equivalent Expression:
The expression [tex]\(\frac{7}{4}\)[/tex] should match one of the given options of logarithms. We can use another property of logarithms, which states:
[tex]\[
\frac{\log_a b}{\log_a c} = \log_c b
\][/tex]
So, the simplified expression [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] can be represented as [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].
1. Understand What the Expression Means:
The expression [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] represents the ratio between [tex]\(\log_2 128\)[/tex] and [tex]\(\log_2 16\)[/tex].
2. Convert Logarithms Using Change of Base:
One important property of logarithms is that if [tex]\(\log_a b = x\)[/tex], then [tex]\(a^x = b\)[/tex].
3. Evaluate [tex]\(\log_2 128\)[/tex] and [tex]\(\log_2 16\)[/tex]:
- First, calculate [tex]\(\log_2 128\)[/tex].
- Since [tex]\(2^7 = 128\)[/tex], it follows that [tex]\(\log_2 128 = 7\)[/tex].
- Now, calculate [tex]\(\log_2 16\)[/tex].
- Since [tex]\(2^4 = 16\)[/tex], it follows that [tex]\(\log_2 16 = 4\)[/tex].
4. Simplify the Fraction:
- Now, substitute the values into the expression:
[tex]\[
\frac{\log_2 128}{\log_2 16} = \frac{7}{4}
\][/tex]
5. Identify the Equivalent Expression:
The expression [tex]\(\frac{7}{4}\)[/tex] should match one of the given options of logarithms. We can use another property of logarithms, which states:
[tex]\[
\frac{\log_a b}{\log_a c} = \log_c b
\][/tex]
So, the simplified expression [tex]\(\frac{\log_2 128}{\log_2 16}\)[/tex] can be represented as [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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