What is the logarithm or root operation in type-space?
A type $C$ has a logarithm to base $X$ of $P$ exactly when $C \cong P\to X$. That is, $C$ can be seen as a container of $X$ elements in positions given by $P$. Indeed, it's a matter of asking to what power $P$ we must raise $X$ to obtain $C$.

It makes sense to work with $\mathop{log}F$ where $F$ is a functor, whenever the logarithm exists, meaning $\mathop{log}\!_X(F\:X)$. Note that if $F\:X\cong \mathop{log}F\to X$, then we certainly have $F\:1\cong 1$, so the container tells us nothing interesting other than its elements: containers with a choice of shapes do not have logarithms.

Familiar laws of logarithms make sense when you think in terms of position sets

$$\begin{array}{rcl@{\qquad}l}
 \mathop{log} (\mathop{K}1) &amp;=&amp; 0   &amp; \mbox{no positions in empty container}\\
 \mathop{log} I &amp;=&amp; 1 &amp; \mbox{container for one, one position}\\
\mathop{log} (F\times G) &amp;=&amp; \mathop{log}F+\mathop{log}G &amp;
  \mbox{pair of containers, choice of positions} \\
\mathop{log} (F\cdot G) &amp;=&amp; \mathop{log}F\times\mathop{log}G &amp;
  \mbox{container of containers, pair of positions}
\end{array}$$

We also gain $\mathop{log}\!_X(\nu Y. T) = \mu Z. \mathop{log}\!_X T$ where $Z=\mathop{log}\!_XY$ under the binder. That is, the path to each element in some codata is defined inductively by iterating the logarithm. E.g.,

$$\mathop{log}\mathop{Stream} = \mathop{log}\!_X(\nu Y. X\times Y) = \mu Z. 1 + Z = \mathop{Nat}$$

Given that the derivative tells us the type in one-hole contexts and the logarithm tells us positions, we should expect a connection, and indeed

$$F\:1\cong 1 \;\Rightarrow\; \mathop{log}F\cong \partial F\:1$$

Where there is no choice of shape, a position is just the same as a one-hole context with the elements rubbed out. More generally, $\partial F\:1$ always represents the choice of an $F$ shape together with an element position within that shape.

I'm afraid I have less to say about roots, but one could start from a similar definition and follow one's nose. For more uses of logarithms of types, check Ralf Hinze's "Memo functions, polytypically!". Gotta run...