how would you define 2 to the power of x (2^x) if x is irrational?

Answer 1

As answered in another question, there are two standard ways:

1. Define 2^x in the usual way for rational exponents. Then, if x is irrational, find a sequence of rational numbers x_n converging to x (the partial decimal expansions always work) and define 2^x as the limit of 2^(x_n). The difficulty with this is showing that the limit exists and does not depend on the specific sequence x_n. Conceptually, it is the easier way. In details it is more complicated.

2. Define either exp(x) as a power series or ln(x) as an integral with the other function being the inverse of the one you define. Then set 2^x=exp(x*ln(2)). The problem with this method is showing consistency with the previous definitions for rational x.

Answer 2

if x is irrational in 2^x, then 2^x could be either irrational or rational.

If x = log base 2 of 5 then x is irrational and 2^x = 5.
if x = 1 then x is rational and 2^x = 2.

Answer 3

2^x is exp(x * log(2)), whether x is rational or irrational. When x is rational, you can write it down in other neater ways, but they're not definitions, so irrational x is still okay.