Plane trigonometry, by S.L. Loney.

LOGARITHMS.

147

135. Logarithm. Def. If a be any number and x
and N two other numbers such that ax = N, then x is called
the logarithm of N to the base a and is written log, N.
Exs. Since 102=100, therefore 2=logo1 100.
Since 105 - 100000, therefore 5 = logo0 100000.
Since 24 16, therefore 4=log2 16.
Since 8[8 2 22= 4, therefore = log8 4.
Since 9- = - = 3 = therefore
9 33 - 27'
3 /1\
-2=lo g27)
N.B. Since a~ = 1 always, the logarithm of unity to any base is always
zero.
136. In Algebra, if m and n be any real quantities
whatever, the following laws, known as the laws of indices,
are found to be true:
(i) am x an = am+n
(ii) am - a = am-n,
and (iii) (am)n = an.
Corresponding to these we have three fundamental
laws of logarithms, viz.
(i) 1oga (mn) = loga mn + log. n,
(ii) l oga (n) = - l oga m n- oga n,
and (iii) log, mn = n loga m.
The proofs of these laws are given in the following
articles.
10-2