[Maths]
Question:- Find the value of infinite tatration.
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Answers
Pre-requisite Knowledge:
Lambert W function
In order to simplify the given tetration, we will be using a special type of function, know as Lambert W function which is also called productlog function.
Lambert W function is defined as where
.
And in general, if we have any equation of the type .
Logarithmic properties
We will use the following logarithmic properties to solve this question.
Evaluation:
Let the given tetration be equal to some variable (let's say y) and we will solve for y.
Taking natural log both sides.
Multiply both sides with -1.
Exponentiating both sides.
Multiply both sides with .
Now, we can see that this is forming general form of Lambert W function.
Therefore, our required result is:
Bonus question:
Let's solve another interesting question involving Lambert W function.
Question:
Solve for x in the equation .
Evaluation:
Taking natural log both sides.
Now this is forming usual definition of Lambert W function, so let's introduce it!
Exponentiating both sides.
This is the value of satisfying the given equation.