Final answer:
The function log n - √n grows asymptotically slower than √n. Thus, the correct asymptotic order of growth for this function is O(√n), which corresponds to option 2 provided in the question.
Step-by-step explanation:
The question asks to select the correct inequality for the asymptotic order of growth of the function log n - √n. First, it is important to understand that asymptotic notations, such as Big O notation, describe the upper bounds of an algorithm's growth rate.
We know from the properties of logarithms that the natural logarithm (ln), which is logarithm to the base e, tells us the power to which e must be raised to get the number. For example, ln(10) is approximately 2.303 because e2.303 ≈ 10.
Now, to compare the growth rates of log n and √n, we need to understand that for large values of n, the square root function grows faster than the logarithmic function. Logarithms grow very slowly compared to most polynomial functions, and a square root is a polynomial function of degree 1/2.
Therefore, as n becomes large, √n will dominate log n. This means that the negative square root will reduce the value of the logarithm but not dominate it in terms of growth rate. Thus, the overall growth rate of log n - √n is dominated by √n.
Therefore, we can conclude that the correct inequality for the asymptotic order of growth of log n - √n is:
log n - √n = O(√n)
This is option 2 in the question.