John Stillwell - Naive Lie Theory.pdf - Index of

7.1 Logarithm and exponential 141
the same terms occur in the expansion of log(e x ),when|e x − 1| < 1, and
their sum is zero because log(e x )=x under these conditions.
Thus log(e X )=X as required.
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The inverse property allows us to derive certain properties of the matrix
logarithm from corresponding properties of the matrix exponential. For
example:
Multiplicative property of matrix logarithm. If AB = BA, and log(A),
log(B), and log(AB) are all defined, then
log(AB)=log(A)+log(B).
Proof. Suppose that log(A)=X and log(B)=Y ,soe X = A and e Y = B by
the inverse property of log. Notice that XY = YX because
(A − 1)2 (A − 1)3
X = log(1 +(A − 1)) = (A − 1) − + −···,
2 3
(B − 1)2 (B − 1)3
Y = log(1 +(B − 1)) = (B − 1) − + −···,
2 3
and the series commute because A and B do. Thus it follows from the
addition formula for exp proved in Section 5.2 that
AB = e X e Y = e X+Y .
Taking log of both sides of this equation, we get
log(AB)=X +Y = log(A)+log(B)
by the inverse property of the matrix logarithm again.
Exercises
The log series
log(1 + x)=x − x2
2 + x3
3 − x4
4 + ···
was first published by Nicholas Mercator in a book entitled Logarithmotechnia in
1668. Mercator’s derivation of the series was essentially this:
∫ x ∫
dt x
log(1 + x)=
0 1 + t = (1 − t + t 2 − t 3 + ···)dt = x − x2
0
2 + x3
3 − x4
4 + ···.
Isaac Newton discovered the log series at about the same time, but took the idea
further, discovering the inverse relationship with the exponential series as well.
He discovered the exponential series by solving the equation y = log(1 + x) as
follows.
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