Answer :

To evaluate [tex]\(\log_6 172\)[/tex] to the nearest hundredth, we can use the change of base formula. This formula allows us to calculate a logarithm with any base by converting it to common logarithms (base 10) or natural logarithms (base [tex]\(e\)[/tex]).

Here are the steps to solve this:

1. Change of Base Formula: The change of base formula states that for any positive number [tex]\(a\)[/tex] and any positive base [tex]\(b\)[/tex] (where [tex]\(b \neq 1\)[/tex], and [tex]\(c \neq 1\)[/tex]), the logarithm of [tex]\(a\)[/tex] to the base [tex]\(b\)[/tex] can be calculated as:
[tex]\[
\log_b a = \frac{\log_c a}{\log_c b}
\][/tex]
Here, [tex]\(c\)[/tex] can be any positive number that is different from 1. Common choices are [tex]\(c = 10\)[/tex] (the common logarithm) or [tex]\(c = e\)[/tex] (the natural logarithm).

2. Applying the Formula: We need to calculate [tex]\(\log_6 172\)[/tex]. Using natural logarithms (base [tex]\(e\)[/tex]), the formula becomes:
[tex]\[
\log_6 172 = \frac{\ln 172}{\ln 6}
\][/tex]

3. Calculate [tex]\(\ln 172\)[/tex] and [tex]\(\ln 6\)[/tex]:
- [tex]\(\ln 172 \approx 5.1475\)[/tex]
- [tex]\(\ln 6 \approx 1.7918\)[/tex]

4. Divide the Results:
[tex]\[
\log_6 172 \approx \frac{5.1475}{1.7918} \approx 2.8729
\][/tex]

5. Round the Result: Finally, round the result to the nearest hundredth:
[tex]\[
\log_6 172 \approx 2.87
\][/tex]

Therefore, [tex]\(\log_6 172\)[/tex] to the nearest hundredth is approximately 2.87.

Other Questions