To determine the value of [tex]\(\log _6 36 + \log _3 27\)[/tex], let's break it down step-by-step.
1. Calculate [tex]\(\log_6 36\)[/tex]:
- The logarithm [tex]\(\log_6 36\)[/tex] asks the question: "To what power must 6 be raised to obtain 36?"
- We know that [tex]\(6^2 = 36\)[/tex].
- Thus, [tex]\(\log_6 36 = 2\)[/tex].
2. Calculate [tex]\(\log_3 27\)[/tex]:
- The logarithm [tex]\(\log_3 27\)[/tex] asks the question: "To what power must 3 be raised to obtain 27?"
- We know that [tex]\(3^3 = 27\)[/tex].
- Thus, [tex]\(\log_3 27 = 3\)[/tex].
3. Add the two results together:
- [tex]\(\log_6 36 + \log_3 27 = 2 + 3 = 5\)[/tex].
Therefore, the value of [tex]\(\log_6 36 + \log_3 27\)[/tex] is [tex]\(\boxed{5}\)[/tex].
