We know that if eˣ = 270, then x = ln(x) = 5.5984.
Threfore the expansion of 270 in exponential form is
eˣ = 1 + x + x²/2! + . . . + xⁿ/n! + . . . = 270
Test the expansion by evaluating the first 10 terms (with the calculator).
n xⁿ/n! Series sum
------ ------------ -------------------
0 1 1
1 5.5984 6.5984
2 15.6712 22.2696
3 29.2446 51.5142
4 40.9309 92.4451
5 45.8297 138.2748
6 42.7623 181.0371
7 34.2002 215.2373
8 23.9334 239.1707
9 14.8877 254.0584
10 8.3348 262.3932
It is clear that the series quickly approaches the value of 270 after 10 terms.
The relative error is (270 - 262.3932)/270 = 0.0282 = 2.8%, which is small.
