The state of modern mathematical notation is rather curious, as can be demonstrated in observing the various shorthand employed in transcribing natural log functions.

ln(x)+ ln(x)= 2 ln(x)

ln(x) * ln(x)= [ln(x)]²

ln 2 + ln(x)= ln (2x)

ln (x)- ln(2) = ln (x/2)

All of the aforementioned are useful methods of conserving time in the written communication of such expressions. Yet perhaps the most cumbersome operation to perform with the natural log function is to take the natural log of a natural log, ln(ln(x)).  This can be done indefinitely, with the expression  ln(ln(ln(ln(ln(ln(ln(ln(ln(ln(ln(ln(x)))))))))))) and its lengthier brethren at least conceivable. Surely, to write such a behemoth of a term more than twice at the most, and especially to consider it in the midst of the remaining complexity of a typical calculus-level problem, will exhaust one’s energies and entrap one in a quagmire of rote busy work prior to the essentials of the problem ever being considered. Surprisingly, however, no shorthand exists for these functions, despite their not-so-rare appearance in common integrals. The mathematical community has left this field almost entirely uncharted and unnamed, except for what it may consider to be chance tangential inroads.

That state of affairs shall linger no more, as I intend to undertake the systematic study that mainstream mathematics has averted itself to.

In mathematics as in philosophy, before any concept can be legitimately considered, it must be named and defined. Hence: 

A Stolyarovian Function shall be defined as any function that is the function of its own function of any quantity, wherein this condition of being a function of a function can be compounded any multiplicity of times.

For any function f(x), f(f(x)) is a Stolyarovian Function. So is f(f(f(x))). So is

f(f(f(f(f(f(f(f(f(x))))))))). This definition is as expansive as we need in order to develop a shorthand notation for the cumbersome expressions now known as Stolyarovian Functions that would render them easily tractable to our aims.

This notation must be sufficiently distinct from already existing numerical subscripts and superscripts that no confusion among them could ever arise. It must also be applicable to and distinctly recognizable in such functions as logarithms of particular bases. Moreover, any symbols employed should be commonplace and accessible on a standard word processor, as well as indicative in itself as to the nature of the function described.

A symbol that would fit all of the above criteria is a star: «. Every man has seen it, any man can draw it, it can easily be compacted into a superscript just over the location where a function’s subscript, if any, could be placed. It possesses no present usage in mathematical notation, and its first two letters, by pleasant coincidence, happen to be identical to the first two letters of “Stolyarovian Function.”

Presently, we proceed to apply the star in our shorthand. When we are given a non-Stolyarovian Function, i.e. f(x), the star is unnecessary; it would merely magnify the amount of notation before us. However, at any point of greater complexity, it condenses the task of writing. Therefore, we can let f(f(x))= f^«x. When a greater quantity of identical functions within each other exists, the star can merely obtain a coefficient before it. For example, we may set f(f(f(x))= f^2«x.

As a general rule, the coefficient before a star is equal to the number of identical functions within the parentheses of the first function f(x) in a Stolyarovian Function.

To witness the immense ease and convenience which this notation imparts upon our formerly encountered behemoths, we observe the elegance attainable with the new system:

ln(ln(x))= ln^«x

ln(ln(ln(ln(ln(ln(ln(ln(ln(ln(ln(ln(x))))))))))))= ln^11«x

Some additional specimens can also be tamed:

sin(sin(sin(x)))= sin^2«x

arcsec(arcsec(arcsec(arcsec(arcsec(x)))))= arcsec^4«x  

log₂(log₂(log₂(log₂(x))))= log₂^3«x

e^(e^(e^x))= e^(^2«x) 

The star system of notation for Stolyarovian Functions also contributes immensely to the mathematical facets of analyzing the properties of said functions. Let us take elementary differentiation as an example. It is a known property that if u=f(x) and y=ln(u), then y’= (1/u)(u’), where u’ is the derivative of u.

If y= ln(x), y’= 1/x. To obtain values of derivatives of Stolyarovian Functions involving natural logarithms, the following table will provide a systematic framework.

It would be a gargantuan task indeed to write the derivative of ln ^4«x without star notation. Moreover, one would need to proceed by opening successively each set of parentheses finding u’ 5 times before reaching an answer which should occupy less than a second of one’s ponderance. By the simple property of derivation of ln functions, one should deduce Property of Stolyarovian Natural Logarithmic Functions #1

(PSNLF #1):
If y= ln ^m«u, where u= f(x),

Then y’= u’/[(u ln u)( ln^«u)(ln ^2«u)… (ln ^m-2«u) ( ln ^m-1«u)].

Once this property is fathomed, the determination of derivatives of ln ^4« (x²+1),

ln ^12«(12x³), and even ln ^77«(sin x) should not require more than an inkling of mental effort. All that is needed is a firm knowledge of the derivatives of u, the function in the innermost parentheses. PSNLF #1 shall do the remainder of the work.

What of integration? Numerous expressions can be integrated if they are a product of two quantities where one is the base of the derivative of the other. Given that principle and PSNLF #1, it is possible to venture into the field of Stolyarovian functions raised to a power. For example:

∫ ln ^4«x/[(x ln x) (ln^«x) (ln ^2«x) (ln ^3«x)]= ∫ ln ^4«x * {1/[(x ln x) (ln^«x) (ln ^2«x) (ln ^3«x)]}=  (ln ^4«x)²/2 + Constant.

Similarly, knowing that if y= uⁿ, then y’= nu^n-1* u’, one can differentiate Stolyarovian Natural Logarithmic Functions raised to some power greater than one. For example:

Y=(ln ^3«x³)^5/7

Y’= [(3x²)(5/7)(ln ^3«x³)^-2/7]/[(x³ ln x³) (ln^«x³) (ln ^2«x³)]

= [15(ln ^3«x³)^-2/7]/[(7x ln x³) (ln^«x³) (ln ^2«x³)]

The result is an immensely ornate and advanced expression that nevertheless needs not be cumbersome. Star notation has excised all superfluity from it, while PSNLF #1 combined with elementary rules of differentiation and knowledge of elementary derivatives can reduce to less than a minute a problem that before may have occupied tens of such time intervals.

The study of Stolyarovian Functions has merely begun, and their further systematization is the task before me in future works. Similar regularities exist in non-natural-logarithmic Stolyarovian Functions, though they may be subtler to spot, due to the nature of the derivatives, for example, of sine, arccotangent, and non-e based log functions. I shall aim to uncover and explore these functions in future treatises. For the present, I have charted new territory and established an essential foothold. The expansion shall begin from there.