# Operating Systems – How can we calculate how many processes should be in memory at the same time to maximize CPU utilization?

To let $$U$$ specify the CPU utilization in percent. What you have calculated is:
$$log (U) = log (1 – P ^ n) stackrel {!} = log (1) – log (P ^ n) = log (1) – n log (P)$$

However, the equality marked with an exclamation mark is incorrect $$log (a – b)$$ is in general Not equal $$log (a) – log (b)$$,

The correct argument restores the original equality as $$P ^ n = 1 – U$$ and then:
$$n log (P) = log (P ^ n) = log (1 – U)$$
Thus:
$$n = frac { log (1 – U)} { log (P)}$$

(Of course, everything provided $$P> 0$$ and $$U <1$$; Otherwise there is no (unambiguous) solution.)

Tip: When you rewrite math expressions, you always do one step at a timeand question every single step! Otherwise you are bound to make mistakes. (Source: personal experience)