I was thinking over the topic and try to put it on its feet.

The paradigm is that shall exist for each .

So if this limit exists, we let ,

(Jay's change of base formula)

,

Change of base is just the application of a function (I think one could show that is analytic):

While does not depend on any or , does. we can define it by setting in the second equation:

.

So we need to show in your case that the limit:

exists for in some initial range, where is your linear approximation.

To show this we could use the Cauchy criterion. It should work in the form exists, if for each there exists a and a such that for all and : .

Where we put .

Suprisingly but happily the compositional difference is independent on :

.

Well, I dont pretend that it helps you, but it helped me at least somewhat in understanding

The paradigm is that shall exist for each .

So if this limit exists, we let ,

(Jay's change of base formula)

,

Change of base is just the application of a function (I think one could show that is analytic):

While does not depend on any or , does. we can define it by setting in the second equation:

.

So we need to show in your case that the limit:

exists for in some initial range, where is your linear approximation.

To show this we could use the Cauchy criterion. It should work in the form exists, if for each there exists a and a such that for all and : .

Where we put .

Suprisingly but happily the compositional difference is independent on :

.

Well, I dont pretend that it helps you, but it helped me at least somewhat in understanding