Question

What value correctly evaluates log7492+log77
using the properties of logarithms?(1 point)
Responses

4
4

5
5

6
6

3

Answer 1

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).