To evaluate \( \log_7 492 + \log_7 7 \), we can use the property of logarithms that states:
\[ \log_b a + \log_b c = \log_b (a \cdot c) \]
In this case, we have:
\[ \log_7 492 + \log_7 7 = \log_7 (492 \cdot 7) \]
Calculating \( 492 \cdot 7 \):
\[ 492 \cdot 7 = 3444 \]
Now, we need to express \( \log_7 3444 \). However, we can simplify \( \log_7 7 \) first:
\[ \log_7 7 = 1 \]
So we have:
\[ \log_7 492 + 1 = \log_7 3444 \]
To find \( \log_7 3444 \), we need to express \( 3444 \) in terms of powers of \( 7 \). Since \( 3444 \) does not equal any integer powers of \( 7 \) directly, and calculating \( \log_7 3444 \) explicitly is not straightforward, we can check for integer values:
To see how \( \log_7 3444 \) evaluates, we might numerically test powers of \( 7 \):
- \( 7^3 = 343 \)
- \( 7^4 = 2401 \)
- \( 7^5 = 16807 \) (too high)
Since \( 3444 \) is between \( 7^4 \) and \( 7^5 \), we can estimate:
- \( 7^4 < 3444 < 7^5 \) implies \( 4 < \log_7 3444 < 5 \).
Therefore, we can conclude that the logarithms do not produce an integer. Since the problem asks for values, the answer corresponding to integer approximation is \( 5 \), but it may indicate a syllabus misdirection if we must provide an explicit logarithmic sum structure directly yielding integers.
To finalize, we see \( \log_7 492 + \log_7 7 = \log_7(3444) \) indicates that summative evaluation implies likely considerations to evaluate final integer results, giving:
Answer: 5 (closer to true estimated evaluation).