I was wandering the following:
Suppose two $\mathbb Q$-valued Cauchy sequences $(a_n)$ and $(b_n)$ are equivalent provided $a_n-b_n \to 0$. We may define $\mathbb R$ as the quotient set of the set of all $\mathbb Q$-valued Cauchy sequences with respect to the above equivalence relation.
Then we may easily define the sums, products and hence rational powers of real numbers.
However, does it make sense to extend the definition to real powers? Namely, suppose $a, b \in \mathbb R$ with $a >0$ and suppose $ \mathbb Q\ni b_n \to b$. Is the following definition
$$a^b := \lim_{n \to \infty} a^{b_n}$$
consistent (i.e. does the limit exist and is the same for all $\mathbb Q \ni b_n \to b$)?
My reason for asking is that I've always seen $a^b$ formally defined as $\exp(b \log a)$ and I was wandering if such machinery is really needed or we can give a more "elementary" definition of real exponentiation without invoking the knowledge of $\exp$ and $\log$.
Thanks in advance!