Help with math calc

Calculating efficiency of loops from both

Gilberg and Hortsmann books...

Logarithmic Loops (multiplication and division):

Where log x (log base x) is the multiple/divisor value

Note with division we flip n and m in the formulas above.

Basically I trying to find an all purpose formula- derived from the following..

A loop where the loop control body is multiplied or divided...

f(n) = ceil(log2 n) or where N is the iterations...

what I came up with

Cases 1 and 2 are applicable to logarithmic loops as well.

3. Valid Execution:

Same test as above except:

for (i = m; i < n; i *= x)

do something

o If the condition is < or > the formula is:

num = ceiling (log x (n / m))

o If the condition is ≤ or ≥ the formula is:

num = floor (log x (n / m)) + 1

Any additional insight for those math inclined with my quest of a general formula..