# Operating systems: How can we calculate how many processes must be in memory at the same time to maximize CPU usage?

Leave $$U$$ Indicates the CPU utilization in percentage. 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 since $$log (a – b)$$ is, in general, do not equal to $$log (a) – log (b)$$.

The correct line of reasoning is rethinking the original equality as $$P ^ n = 1 – U$$ and so:
$$n log (P) = log (P ^ n) = log (1 – U)$$
A) Yes:
$$n = frac { log (1 – U)} { log (P)}$$

(Of course, all this assuming $$P> 0$$ Y $$U <1$$; otherwise, there is no (only) solution.

Tip: when rephrasing mathematical expressions, always do it. one step at a time, and question every step! Otherwise, you are bound to make mistakes. (Source: personal experience)