The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form.

Is there possible an extension of real/complex numbers in which logarithms and inverse trigonometric functions can be expressed in terms of exponentials/trigonometric functions and vice versa in closed form?

P.S. I have asked [here][1] but is seems people there just do not understand the question.

What I am talking about is something like this:
$$\frac1\pi\ln \left(\frac{w-\frac{z}{\pi }}{w-1+\frac{z}{\pi }}\right)=\frac1z\cos (2wz)$$

or this:

$$\ln(\sin(w + z)) = 1/z \cos(2 w z)$$

Where $w$ is some element of the extended field, not a complex number. Is this possible?

  [1]: http://math.stackexchange.com/questions/1769459/can-there-be-a-numerical-system-in-which-logarithms-can-be-expressed-in-terms-of?noredirect=1#comment3611127_1769459