Certainly! Let's break down the given expression and compare it with all the provided options step by step.
### Given Expression:
[tex]\[
\log_3 3 + \log_3 27
\][/tex]
### Step-by-Step Analysis:
1. Simplify the Expression:
- Recall that the logarithmic identity [tex]\(\log_a(a) = 1\)[/tex].
[tex]\[
\log_3 3 = 1
\][/tex]
- Next, use the properties of logarithms. Remember that [tex]\(\log_b(mn) = \log_b(m) + \log_b(n) \)[/tex].
[tex]\[
\log_3 27 = \log_3 (3^3)
\][/tex]
Using the power rule for logarithms [tex]\(\log_b(m^n) = n \log_b(m)\)[/tex]:
[tex]\[
\log_3 (3^3) = 3 \log_3 3 = 3 \cdot 1 = 3
\][/tex]
- Add the two simplified parts:
[tex]\[
\log_3 3 + \log_3 27 = 1 + 3 = 4
\][/tex]
### Comparison with Options:
Let's now evaluate each provided option.
Option A: [tex]\(\log_3 81\)[/tex]
[tex]\[
\log_3 81 = \log_3 (3^4)
\][/tex]
Using the power rule:
[tex]\[
\log_3 (3^4) = 4 \log_3 3 = 4 \cdot 1 = 4
\][/tex]
Since [tex]\(\log_3 3 + \log_3 27 = 4\)[/tex] and [tex]\(\log_3 81 = 4\)[/tex], they are equivalent.
Option B: [tex]\(\log_3(3^4)\)[/tex]
[tex]\[
\log_3(3^4) = 4 \log_3 3 = 4 \cdot 1 = 4
\][/tex]
Since [tex]\(\log_3 3 + \log_3 27 = 4\)[/tex] and [tex]\(\log_3(3^4) = 4\)[/tex], they are equivalent.
Option C: 4
We already found that [tex]\(\log_3 3 + \log_3 27 = 4\)[/tex]. So, this option is indeed equivalent.
Option D: [tex]\(\log 10\)[/tex]
[tex]\(\log 10\)[/tex] is the logarithm to base 10, commonly denoted [tex]\( \log_{10} 10 \)[/tex]. This simplifies to:
[tex]\[
\log_{10} 10 = 1
\][/tex]
Since [tex]\(\log_3 3 + \log_3 27 = 4\)[/tex] and [tex]\(\log 10 = 1\)[/tex], they are not equivalent.
### Conclusion:
Based on the analysis, the expressions equivalent to [tex]\(\log_3 3 + \log_3 27\)[/tex] are:
- A. [tex]\(\log_3 81\)[/tex]
- B. [tex]\(\log_3(3^4)\)[/tex]
- C. 4
Thus, the checked answers for the question are:
- A. [tex]\(\log_3 81\)[/tex]
- B. [tex]\(\log_3 (3^4)\)[/tex]
- C. 4
