How to properly calculate the order of growth of an exponential algorithm?
In this answer we assume the arithmetic model of computation, in which each arithmetic operation has cost 1.

The question has bad notation. Define function expt b n to compute b^n as:

expt b 0 = 1
expt b n = b ⋅ expt b (n − 1)


Computing expt b n requires computing n-1, computing expt b (n-1), and multiplying that expression by b. So the cost of expt b n is $2 +$ the cost of expt b (n-1). expt b 0 has cost 0.

Putting the above together, we see the cost of exp b n is $2 +$ the cost of exp b (n-1), $= 4 +$ the cost of exp b (n-2), $\ldots$, $= 2n +$ the cost of exp b 0, $= 2n$.