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Answer :
To find the approximate value of [tex]\(\log_4 128\)[/tex] using the given values [tex]\(\log 128 \approx 2.1\)[/tex] and [tex]\(\log 4 \approx 0.6\)[/tex], we can apply the change of base formula. Here are the steps:
1. Understand the Change of Base Formula:
The change of base formula states that for any positive numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (with [tex]\(a, b \neq 1\)[/tex]), the logarithm [tex]\(\log_b a\)[/tex] can be calculated using:
[tex]\[
\log_b a = \frac{\log_c a}{\log_c b}
\][/tex]
In this case, we want [tex]\(\log_4 128\)[/tex]. We can use base 10 (common logarithm) to apply the formula:
[tex]\[
\log_4 128 = \frac{\log_{10} 128}{\log_{10} 4}
\][/tex]
2. Substitute the Given Values:
We have [tex]\(\log_{10} 128 \approx 2.1\)[/tex] and [tex]\(\log_{10} 4 \approx 0.6\)[/tex]. Substitute these into the formula:
[tex]\[
\log_4 128 = \frac{2.1}{0.6}
\][/tex]
3. Perform the Division:
Calculate the division:
[tex]\[
\frac{2.1}{0.6} \approx 3.5
\][/tex]
So, the approximate value of [tex]\(\log_4 128\)[/tex] is [tex]\(3.5\)[/tex].
The solution is [tex]\(3.5\)[/tex].
1. Understand the Change of Base Formula:
The change of base formula states that for any positive numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (with [tex]\(a, b \neq 1\)[/tex]), the logarithm [tex]\(\log_b a\)[/tex] can be calculated using:
[tex]\[
\log_b a = \frac{\log_c a}{\log_c b}
\][/tex]
In this case, we want [tex]\(\log_4 128\)[/tex]. We can use base 10 (common logarithm) to apply the formula:
[tex]\[
\log_4 128 = \frac{\log_{10} 128}{\log_{10} 4}
\][/tex]
2. Substitute the Given Values:
We have [tex]\(\log_{10} 128 \approx 2.1\)[/tex] and [tex]\(\log_{10} 4 \approx 0.6\)[/tex]. Substitute these into the formula:
[tex]\[
\log_4 128 = \frac{2.1}{0.6}
\][/tex]
3. Perform the Division:
Calculate the division:
[tex]\[
\frac{2.1}{0.6} \approx 3.5
\][/tex]
So, the approximate value of [tex]\(\log_4 128\)[/tex] is [tex]\(3.5\)[/tex].
The solution is [tex]\(3.5\)[/tex].
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