Curious Mathematical Problem

Let us consider the following equation:

Prove that

• x = log(Pi) = 1.14472988584... is a very good approximation of a solution, up to 10 digits.
• Using high performance computing or other means, prove that it is correct up to 1,000 digits.
• Is x = log(Pi) an exact solution?

If the answer to the last question is positive, this would mean that log(Pi) is NOT a transcendental number, but rather, an algebraic number. A remarkable result in itself!

Source for picture: algebraic numbers

Solution

Any real number larger or equal to 1 is a solution, so there is nothing particular with log(Pi).

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