To find the number of people infected with the flu virus after 5 days, we substitute $t = 5$ into the given formula:

$N = 3000 - 500 \log(8t + 1)$

Substituting $t = 5$:

$N = 3000 - 500 \log(8(5) + 1)$

Simplify inside the logarithm:

$N = 3000 - 500 \log(40 + 1) = 3000 - 500 \log(41)$

Now, let's calculate the logarithm value:

$\log(41) \approx 1.613$

So,

$N = 3000 - 500 \times 1.613 = 3000 - 806.5$

Finally,

$N \approx 2193.5$

Therefore, approximately 2194 people are infected after 5 days.

Would you like to know more details or have any questions?

Here are 5 related questions:

1. What is the rate of change of the number of infected people with respect to time?
2. How many people were infected on the day the vaccine was introduced (i.e., at $t = 0$)?
3. What will be the number of people infected after 10 days?
4. How does changing the initial number of infected people affect the model?
5. Can you find the time when the number of infected people will be reduced to 2000?

Tip: When dealing with logarithmic functions, it's important to remember their domain restrictions to avoid errors in calculations.