A naïve definition of exponentiation is that, it is just a 'repeated multiplication'. In this sense, $b$ in $a{\wedge}b$ should be an integer. Then we introduced
${\displaystyle \exp(x):=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{24}}+\cdots }$ and ${\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}x^{k}}{k}}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots }$
With the help of these, we can extend the above $b$ to real numbers as $a{\wedge}b:=exp(b{\cdot}ln(a))$
On a similar note, a naïve definition of multiplication is that, it is a 'repeated addition'. In this sense, $b$ in $a*b$ should be an integer. Now, with only the knowledge of addition & subtraction of two real numbers and multiplication of a real number with an integer, how can I arrive at multiplication of two real numbers?
