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Use the values [tex]\log 128 \approx 2.1[/tex] and [tex]\log 4 \approx 0.6[/tex] to find the approximate value of [tex]\log_4 128[/tex].

[tex]\log_4 128 \approx[/tex] ________

The solution is:

Answer :

To find the approximate value of [tex]\(\log_4 128\)[/tex], we can use the change of base formula. Here’s how you do it step-by-step:

1. Understand the Change of Base Formula: This formula allows us to change the base of a logarithm. It is:
[tex]\[
\log_b a = \frac{\log_c a}{\log_c b}
\][/tex]
where [tex]\(b\)[/tex] is the new base, [tex]\(a\)[/tex] is the number you are taking the logarithm of, and [tex]\(c\)[/tex] is any common base (often 10 or [tex]\(e\)[/tex]).

2. Identify the Given Values:
- [tex]\(\log 128 \approx 2.1\)[/tex]
- [tex]\(\log 4 \approx 0.6\)[/tex]

3. Apply the Change of Base Formula:
- We want to find [tex]\(\log_4 128\)[/tex].
- Using the change of base formula, we can write:
[tex]\[
\log_4 128 = \frac{\log 128}{\log 4}
\][/tex]

4. Substitute the Given Values:
- Substitute [tex]\(\log 128 \approx 2.1\)[/tex] and [tex]\(\log 4 \approx 0.6\)[/tex] into the formula:
[tex]\[
\log_4 128 = \frac{2.1}{0.6}
\][/tex]

5. Calculate the Result:
- Performing the division gives:
[tex]\[
\log_4 128 \approx 3.5
\][/tex]

So, the approximate value of [tex]\(\log_4 128\)[/tex] is 3.5.

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