$$\text{Let }\log^2(x)=\log(\log(x)),\\ \text{ then }f(y,x)=\log^{\lfloor1+y\rfloor}\left(\log(x)/\log((1-x^{1/x}(y-\lfloor y\rfloor))+(y-\lfloor y\rfloor))\right)$$
gives an interpolation between $\log^{\lfloor y\rfloor}(x)\text{ and }\log^{\lceil y\rceil}(x).$
log[y_, x_] := N[NestList[Log,Log[x]/Log[(1-FractionalPart[y]) x^(1/x)+
FractionalPart[y] E], Floor[y + 1]][[Floor[y + 1]]]]
As mike4ty4 says in his answer here,
there isn't a unique way to interpolate the values of tetration at integer "height"
and presumably the same goes for logarithmic interpolation. Is there a generally accepted standard method for interpolating log functions? I looked at the answer to this question, but I am not any clearer on this particular issue.
