The basic intuition for this is to start from the origin and make small changes to the function based on its known derivative at the origin.
More precisely, we know that for any base b, exponentiation satisfies:
And we also know that for in particular that we satisfy the exponential function differential equation and so: exponential function differential equation is the value around , which is quite little information! This idea is basically what is behind the importance of the ralationship between Lie group-Lie algebra correspondence via the exponential map. In the more general settings of groups and manifolds, restricting ourselves to be near the origin is a huge advantage.
Now suppose that we want to calculate . The idea is to start from and then then to use the first order of the Taylor series to extend the known value of to .
- Exponential map | 130