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[tex]<x'|e^{i\hat{x}}|x>=e^{ix'}\delta(x'-x) [/tex]

where where [tex]\hat{x}[/tex] is an operator?

I mean if

[tex]\hat{x}|x>=x|x> [/tex]

we may write [tex]<x'|\hat{x}|x>=x<x'|x>=x\delta(x'-x)[/tex]

but in the expression at the top, we have an exponential operator (something I've never come across before) - is |x> an eigenstate of this operator (it seems to be), and why is the eigenvalue of the operator exactly the same form as the operator?

Thanks.