-1

If the identity number of the addition/subtraction pair of operations is 0.

and the identity number of the multiplication/division pair of operations is 1.

what is the identity number of the exponentiation/logarithm pair of operations?

honestSalami
  • 245
  • 2
    Do you mean to consider $a^b$ with $^$ as your operation and $a$ and $b$ as the inputs? If so, what makes you think this should have an identity? – Randall Oct 18 '21 at 16:08
  • 3
    Also, subtraction does not have an identity, nor does division (at least not two-sided). – Randall Oct 18 '21 at 16:11
  • well, I don't know if it has an identity. And I don't know how to find out. That's why I'm asking – honestSalami Oct 18 '21 at 16:13
  • what do you mean by a two sided identity? – honestSalami Oct 18 '21 at 16:13
  • 2
    Assuming you are working with real numbers (this isn't clear), there is no element $a$ such that $x-a=x$ and $a-x=x$ for all $x$. You only get it on one side, which is $x-a=x$ for all $x$ (with the winner being $a=0$, obviously). – Randall Oct 18 '21 at 16:15
  • 2
    If $a^b=a$ then b=1. but $a^b=b$, then a is not a constant. – MH.Lee Oct 18 '21 at 16:16
  • I see, when you consider subtraction as an operation, there really is no identity. I was considering subtraction as a special case of addition by a negative number – honestSalami Oct 18 '21 at 16:16
  • Problems are that exponentiation is not commutative i.e $a^b\neq b^a$ so there's only one side identity i.e $a^1=a$. While for the other side there's no identity that works for all real numbers in general. Exponentiation in terms of properties is pretty poor, no associativity, no commutivity etc. – kingW3 Oct 18 '21 at 17:16
  • @kingW3 yeah! I was wondering if there might be an identity in another realm besides the real numbers? maybe in the complex numbers, or the surreal numbers? – honestSalami Oct 18 '21 at 22:14

1 Answers1

1

From the discussion above, its clear that exponentiation does not have an identity in the real numbers.

Wikipedia says that there are left identities (where e*a=a) and right identities (where a*e=a), and that if a single number is both its called a two sided identity,and is accepted as the identity element for that operation.

The problem with exponentiation is that there is only a right identity (a^1=a) and there is no left identity. So while you might be tempted to call 1 the identity of exponentiation, its just on one side and does not count for both.

So, at least in the real numbers, exponentiation does not have an identity.

See this answer for a clear explanation of why its a mistake to assume that exponentiation only has one inverse operation (the other is rooting)

And this for a little tour on MAKING an operation like exponentiation that has an identity.

honestSalami
  • 245