Equational theory of $\varepsilon_0$ under the signature $(+,*,exp,0,1)$

Question

I want to know if there is a finite set of identities that characterize the equational theory of the ordinal $\varepsilon_0$ under addition, multiplication, exponentiation, $0$, and $1$. $\varepsilon_0$ is understood to be the set of ordinals less than $\varepsilon_0$. I conjecture that these identities are sufficient:

1. $x+0=x$
2. $0+x=x$
3. $(x+y)+z=x+(y+z)$
4. $x*0=0$
5. $0*x=0$
6. $x*1=x$
7. $1*x=x$
8. $(x*y)*z=x*(y*z)$
9. $x*(y+z)=(x*y)+(x*z)$
10. $x^0=1$
11. $x^1=x$
12. $1^x=1$
13. $x^y*x^z=x^{y+z}$
14. $(x^y)^z=x^{y*z}$

Are these sufficient, or do we need a few more identities? Or is it not in fact finitely axiomatizable?