Details about the function n/ln(n)

[tuhinrao]

From the file attached I would like to know the following.
if Y=n/ln(n), Is there a way of explicitly expressing n in terms of Y.


Relations I found are:

There are 2 values of n for every Y. Except at Y=e , the two values converge to n=e.
If n1 and n2 are the values of n
then
n1^n2=n2^n1.

So is there a way of finding n1, given n2?

What could be the possible type of functions involved?
From the graph it is seen that |n1-e| and |n2-e| are related inversely.
What could be this relation?

 

[haruspex]

No, there's no way to express n as a function of y using any standard functions. Of course, it's always possible to invent a new one for the purpose.
Mixtures functions of different types (polynomial, exponential, logarithmic, trigonometric..) are nearly always impossible to invert. E.g. y = x.exp(x), y = sin(x)/x, ...

 

[Mute]

There is a special function called the Lambert-W function (aka the Product-Log) which you can use to write n in terms of y. The Lambert-W function is the function w=W(x) such that
x = w exp(w).

If you invert your equation so that 1/y = ln(n)/n, and then let n = exp(a), this gives

$$\frac{1}{y} = a e^{-a}$$

We see that if we multiply both sides by -1 this will be in Lambert-W form, giving a = W(-1/y). Inverting n = exp(a), this gives

$$n = \ln W_k\left(-\frac{1}{y}\right).$$

Some very important notes: The Lambert-W function has two real-valued branches, corresponding to k = 0 and k = -1. Usually the k=0 branch corresponds to the desired solution. The other branches gives complex values for W, so inverting ln n = W(-1/y) is more complicated if want solutions for these others branches. However, since you seem mostly interested in real values you don't need to worry about this.

 

[tuhinrao]

Thanks 'Mute'. Got my way through.

tuhinrao

 

[CellCoree]

I would like to clarify that the function n/ln(n) is known as the logarithmic integral function. It is a special function that is commonly used in mathematics and engineering. It is defined as the integral of 1/ln(x) from 2 to n, where n is a positive real number.

Regarding the question about explicitly expressing n in terms of Y, it is not possible to find a single value of n in terms of Y as there are two possible values of n for every Y, except at Y=e where the two values converge to n=e. This behavior is due to the logarithmic nature of the function, where the values of n increase exponentially as Y increases.

The relation n1^n2=n2^n1 is known as the power property of logarithms and it holds true for any base of logarithm. This relation is also applicable for the logarithmic integral function, as seen in the graph provided. However, it is not possible to find n1 given n2, as there are two possible values of n for every Y.

In terms of the type of functions involved, the logarithmic integral function can be considered as a special type of logarithmic function. It is also related to the natural logarithm function, ln(x), as it is the inverse function of the natural logarithm.

As for the relation between |n1-e| and |n2-e|, it is an inverse relationship as seen in the graph. This can be explained by the fact that as Y increases, the values of n also increase, but they approach the value of e. This means that as n1 and n2 get farther away from e, their difference decreases.

In conclusion, the function n/ln(n) is a special mathematical function that is commonly used in various fields. It is not possible to explicitly express n in terms of Y, but there are certain properties and relations that hold true for this function.