How to maximize y=(c/x)^x using logarithmic differentiation

The actual question is: Let C be a large number. Divide it into n equal parts: C/n.
Multiply together all those parts. what value should n be so that this product is maximized? (hint:You will need to wait until we consider logarithmic differentiation). My teacher gave me y=(c/x)^x but I don't know where to go from here. If you could help that would be awesome!

Find when dy/dx = 0

y = (c/x)^x
ln(y) = x * ln(c/x)
ln(y) = x * ln(c) - x * ln(x)
dy/y = ln(c) * dx - x * dx/x - ln(x) * dx
dy/y = dx * (ln(c) - 1 - ln(x))
dy/dx = y * (ln(c) - 1 - ln(x))
dy/dx = y * (ln(c/x) - 1)

dy/dx = 0

y = 0
(c/x)^x = 0
This never happens

ln(c/x) - 1 = 0
ln(c/x) = 1
c/x = e^1
c/x = e
c = x * e
x = c/e

What?