What is the exponential inverse? And first let me explain what I mean when I am talking about inverse.(adsbygoogle = window.adsbygoogle || []).push({});

I am not talking about anything where f(f^{-1}(x)) = x

What I mean is in one operation, when that operation is applied onto the inverse of that operation it equals the identity element of that operation.

For example, 5 + -5 = 0 because 0 is the identity element in addition (x+0=x), and negative is the inverse.

Also for example, 5 * (1/5) = 1 because 1 is the identity element in multiplication (x*1=x), and reciprocal or 1/x is the inverse.

In matrices, [matrix] * [inverse of that matrix] = [identity matrix]

So, what is this for exponentiation?

Let me define what I want to use for identity element: anything, that when that operation is applied to it, it equals the original number. So in exponentiation, the identity element is 1 because x^1 = x;

So now, we want x^what = identity element; And since the identity element of x is 1 then this can become: x^what = 1?

I have considered 0 as the answer, but that doesn't make any sense. Even though x^0 = 1, and thus the identity element, how is 0 the exponential inverse of x? All of the other inverses for addition and multiplication (-x, 1/x) all include x. So why, here, is 0 the inverse; what makes exponentiation and 0 so special?

In addition, root and logarithm do not work. Root doesn't work because x^(x^(1/anything)) does not equal 1 (unless x = 1). and neither does x^log_{x}(anything)

So x^what = 1, where "what" can't be zero, a root, or a logarithm.

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# Inverse of exponentiation

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