mathsman, we have met before, it took less than 30 seconds to dismiss your messages...

In fact, I have NOT proceeded at all in the manner described by you...I do not think in the lines of classical mathematics, on the contrary, I invent my own formulas.

BiJane, my global logarithm formula should be added immediately to the wikipedia page on logarithms, and also it should be included in the algebra/calculus text in high schools...

How did I proceed?

I started with the global formula for the cosine function, using geometry.

Then I obtained the arccos function (the inverse function).

From there, I got the hyperbolic cosine global formula, and of course the global exponential function.

Then, I calculated the inverse function for the hyperbolic cosine:

That is, the nested square root formula.

Remember, I did not use calculus at all, nor in way, shape or form the formula for the exponential function mentioned by you, mathsman.

Afterwards, I summed the nested square root parentheses, to obtain the final formula.

It is the perfect logarithm formula: it connects algebraic functions with elementary transcendental functions, and it makes superfluous the atrocious classical calculation of a logarithm using a characteristic and a mantissa, forced upon all high school students.