How did feynman compute $e^x$ with the accuracy he wanted? Excerpt from the book Surely You're Joking, Mister. Feynman!

Here Feynman calculates $e$ to a couple of powers.
I understand that he luckily knew a couple of logs by heart. What I don't understand is the part where he adjusts the numbers to the accuracy he wanted to have. The accuracy he got in the first calculation e^3.3 was 27.1126.
In the first example it could be : $$(e^{2.3026} \times e) / e ^ {0.0026} = e^{3.3}$$ 
But then he still had to use the power series for the $e^{0.3026}$ AND divide by it afterwards and I can't see how even Feynman would be able to do that.
Maybe by a linear approximation? Thanks in advance.

Regarding the adjustment - don't forget that the derivative of the exponential function is itself. So you have your first approximate answer, and that's good enough to use as the derivative to step to your refined answer.
So $e^{3.3}$ is about $27.1828$, but that's actually $e^{3.3026}$, so we want to adjust down by $0.0026\times 27.1828 \approx 0.07$ so call it $27.11$.
