In a book I'm reading an exponential equation is defined as an equation containing terms of the form a^x (a to the power of x), where x is a real number, a is a real number, a > 0 and a is not equal to 1.

Now I understand why the definition excludes a < 0 (there are many values of x for which a^x is undefined if a < 0, for example if a = -2 and x = 1/2, a^x = (-2)^1/2 = sqrt(-2) which is undefined.)

But I can't understand why the definition excludes a = 0 and a = 1. The following reasons are given in the book (which I don't understand... reasons are given).

If a = 0 then a^x becomes 0^x. This is equal to the constant 0 for all x except x = 0. It is undefined for x = 0. Thus when a = 0, a^x reduces to a constant or is undefined.

Reason why I don't understand the above: Even if a = 0 in a^x = a^y, surely x still equals y.

If a = 1, then a^x = 1^x = 1 for all real values of x. In this case a^x reduces to the constant 1.

Reason why I don't understand the above: Again, even if a = 1 in a^x = a^y, surely x still equals y.

Now I have an idea of my own on why a = 0 and a = 1 are excluded in the definition and I would appreciate your input on the idea. My idea is that if a = 0 or a = 1 the equation is true for all values of x.