What is the exponential inverse? And first let me explain what I mean when I am talking about inverse. 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^logx(anything) So x^what = 1, where "what" can't be zero, a root, or a logarithm.