To determine how many times higher the number of library visits before examination day is compared to the average weekly visits, we need to divide the number of visits during the one-week period by the average number of visits per week.
The number of visits one week before the examination is \( 2 \times 10^3 \) (which is 2000), and the average number of visits per week is \( 5 \times 10^2 \) (which is 500).
Now, we can perform the calculation:
\[ \text{Number of Times Higher} = \frac{2 \times 10^3}{5 \times 10^2} \]
Calculating the fraction:
\[ = \frac{2000}{500} = 4 \]
Thus, the number of library visits before examination day is 4 times higher than the average number of visits in a week.