How to check if this limit is correct or not?

[Kumar8434]

I can't prove it and I've got it by some intuition because not many properties of superlogarithms are known. I don't think anyone can prove it but is there some way to at least check if it is correct.
The limit is:
$$\lim_{h\rightarrow0}slog_{[log_xx+h]}[log_{f(x)}f(x+h)]$$
where $slog$ is the super-logarithm.
What I need to check is if this limit equals:
$$log_{log_xf(x)}[\frac{d(f(x))}{dx}\frac{x}{y}]$$
for any function $f(x)$. It's true when $f(x)=x^n$. But is there some way to check if it's right or wrong for other functions?

 

[Kumar8434]

I can't find any option to edit my above post, so I'm adding this here.
I think there's one more possibility of what that limit might evaluate to. It is $$\frac{d(f(x))}{dx}*\frac{x}{y}*\frac{logx}{logy}$$. It's also satisfied by $f(x)=x^n$ but I can't check it for other functions. Please tell me some way to check which of these expressions should be the correct value of the limit or if both of these are wrong.

 

[mfb]

How do you define the super-logarithm for non-integer values?

 

[Kumar8434]

How do you define the super-logarithm for non-integer values?

I could be wrong but isn't the superlogarithm defined for special types of fractional values just like the super-root?
For example, when
$$^nx=y$$ when n is an integer then,
$$x=^{\frac{1}{n}}y$$
So, $$slog_yx=\frac{1}{n}$$
I think that extends its range to an infinite number of fractional values of the form $\frac{1}{n}$ the interval [0,1]. That makes the graph almost continuous in the interval [0,1]. I need to know that if the limit I've evaluated is in agreement with this almost continuous graph.
And if that's not possible, then use some extension of superlogarithm on reals, if they don't give too much different values.

 

[mfb]

Wikipedia discusses two definitions. Looks like there is no obvious single definition around.

 

[Kumar8434]

Wikipedia discusses two definitions. Looks like there is no obvious single definition around.

Then is it correct if the graph of slogx is assumed to be straight line between the points on which it is defined?