Plane trigonometry, by S.L. Loney.

392 TRIGONOMETRY.
339. It could now be shewn that the general values
of the logarithms of complex quantities satisfy the
ordinary laws of logarithms, viz.
Log mn = Log m + Log n,
and Log = Log m - Log n.
It could also be shewn that Log inl = n Log m + 2p7ri,
where p is some integer or zero. The proof is left as an
exercise for the student.
EXAMPLES. LVIII.
Prove that
1. ai=e-2m {cos (log a)+ i sin (log a)}.
2 2
2. ia=cos {(2m+2) 7rat +i sin {(2m+-) -raV.
3. ii= cos 0 + i sin 0, where
0=(2m+ r. e-2+).
4. If iii'ad inf= A + Bi, principal values only being considered, prove
that
tan - = B, and A2 +B2=e-rB.
5. If ia+i=a+/i, prove that
a2 +g2= e-(4n+l) 7T
6. If (1+ i) +.a +i, prove that one value of tan-1 is
a
pr + q log, 2.
7. If (a + bi)p=mx+Yi, prove that one of the values of Y is
2 tan-'b
a
loge (a2+ b2)'