Measurement 61Suppose we want to fi nd the natural logarithm of 2 (written ln 2). The base of natural
logarithms ise 2.7188281828. Then
log
log.
.
.
2 0 30103
0 43425
0 69315
eTo verify this result,
e0 69315. 2.T o fi nd log 2 of 26,
log
log.
.
.
26
2
1 41497
0 30103
4 70044
The general case is
log
log(^10) log
10
base
of the number
of the base
of the number.. (2.38)
2.17 Semitone Intervals ...................................................................................................
Suppose that we need^122 (the semitone interval in music). We could write
log (^2) log
12
^122. (2.39)
Therefore
10 10
10
1 05946
22
12 03012
0 0250812log.
.
..
This is the same as multiplying 1.05946 by itself 12 times to obtain 2.
1 0 0.02508 is called the antilog of 0.02508. The antilog is also written as log ^1 , antilog 10,
or 10 exp. All these terms mean exactly the same thing.