In this paper we give a general theorem on the linear independence measure of logarithms of rational numbers and, in particular, the linear independence measure of
1
,
log
2
,
log
3
,
log
5
1,\log 2, \log 3, \log 5
and of
1
,
log
2
,
log
3
,
log
5
,
log
7
1,\log 2, \log 3, \log 5, \log 7
. We also give a method to search for polynomials of smallest norm on a real interval
[
a
,
b
]
[a,b]
which may be suitable for computing or improving the linear independence measure of logarithms of rational numbers.