222
FINNEY,D. J., HAZLEWOOD,
T. AND SMITH,
M. J. (1955). J . gen. MiCrobiol. 12,
222-225
Logarithms to Base 2
BY D. J. FI”EY,* TAMARA HAZLEWOOD
AND
MARIAN J:SMITH
Lecturership in the Design and Analysis of Scientific Ezperiment,
University of Ozford
Graphical representation of the growth of a bacterial culture is usually best
achieved by using a logarithmic scale for whatever measurement of the
density of the culture is made. Moreover, as Monod (1949) has pointed out,
both diagrams and associated calculations are more readily interpreted if
logarithms to base 2 are used instead of common logarithms (i.e. base 10): an
increase of 1unit on the logarithmic scale then corresponds to the equivalent
of one division by each of the cells present at the beginning of the interval.
Rather surprisingly, none of the better-known collections of mathematical
tables nor any of the usual sources for miscellaneous tables appears to contain
logarithms to base 2. Of course, for any particular number, multiplication
of its common logarithm by 3-32193(=log, 10) gives the logarithm to base 2,
but performance of this multiplication for many series of data is tedious and
time consuming. As the result of a suggestion from Dr W. E. van Heyningen,
the following table has therefore been computed.? Only 3 places of decimals
are shown, since graphical work will certainly not permit greater accuracy
and even for arithmetical analysis observational data rarely justify the use
of more digits; the table is designed for simple rapid use by microbiologists,
not for serious mathematical purposes.
The table shows the logarithm of every integer from 100 to 999. Thus, by
direct reading,
10gl673= 9.394.
If the highest and lowest densities of a series do not differ by more than a
factor of 10, there may be no necessity to construct any logarithms outside
the range of the table. For example, if the densities range from 1.7 to 8-9,
expression of density in terms of a volume unit 100 times that of the standard
will give values ranging from 170 to 890; all logarithms can then be read
directly from the table and graphed as they stand.
If the densities extend over a wider range, logarithms of numbers not in
the table will be needed. These are obtained by the ‘usual rule that the
logarithm of a product or quotient is the sum or difference of the logarithms
of the two parts. Thus
log, 2692 =log, 4 +log, 673
= 2.000 + 9.394 = 11.394,
log, 0.075 =log, 300 -log, 400 -log, 10
=8.229 - 8.644 -3.322 = -3.737.
* Present address: University of Aberdeen.
t The table was obtained by multiplication of appropriate entries in the US.National
Bureau of Standards Table of Natural Logarithms by log, e and independently checked by
multiplication of entries in Chamber’s Mathematical Tables (Seven Figure Logarithms) by
log, 10.
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Logarithms to base 2
223
When suitable factors are not obvious, powers of 10 can be used, remembering
that log, 10=3-3219,log, 100=6.6439, log, 1000=9.9658; logarithms of
powers of 10 up to 1O1O are appended to the table. Hence
log, 0.875 =log, 875 -log, 1000
=9.773- 9.966= - 0.193.
(Since 0.875 = 7/8,the same value can be obtained as log, 700 -log, 800.)
The measurements will rarely be accurate to more than 3 significant digits,
but, if required, logarithms of quantities intermediate between tabular
values can be derived by the common-sense application of ‘linear interpolation’; thus log, 2692 can alternatively be calculated as at 2/10of the distance
between
log, 2690 = 8.071 +3.322 = 11.393,
log, 2700=8.077 +3.322=11.399,
and
and is 11.394 as before.
REFERENCE
MONOD,J. (1949). The p w t h of bacterial cultures. Annu. Rev. Microbial. 3, 371.
(Received 13 September 1954)
15-2
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224
D. J . Finney, T.Hazlewood and M . J . Smith
Logarithms to base 2
100
110
120
130
14lO
0
6.644
*781
407
7.022
*129
4
1
7
8
3
5
6.
2
6.658 6.672 6.687 6.700 6.714 6.728 6.741 6.755
-870
0883
-833
-794
-845
-820
,807
458
477
a989 7.000
*954
-919
,966
-943
9 31
0109
7.033 7.0U
7.055 7.066 7.077 7-087 7-098
*209
-170
-180
*160
-200
-150
0140
,190
9
6,768
-895
7.011
-119
,219
150
160
170
180
190
7.229
a322
7.267 7.276 7.285
-358
-366
-375
. - 4 3 0451 -459
*524
0531
-539
0607
a615
-600
7.295 7.304 7.313
-384
-392
401
0476
484
-468
-547
-555
-562
.622
-629
-637
200
210
220
230
240
7.644 7-651 7.658 7.665 7-672 7.679 7.687
-721
,714
-741
-755
0728
*748
-735
-781
-814
420
-794
-788
-801
-807
-845 -852
*877
458
-870
4383
4364
-919
-913
-907
-943
*937
-925
-931
7.693 7.700 7.707
-762
-768
*775
-827
433
-839
-895
,889
.go1
-954
-948
-960
250
260
270
280
290
7.966
8.022
-077
-129
-180
8-006 8,011 8.017
*066
-061
*Or1
el19
-114
-124
-170
*175
-165
-219
-214
-224
300
310
320
330
340
8.229 8.234
-276
-281
-322 a326
~366 .371
-409
,414
8.238
*285
-331
9375
-418
350
360
370
380
390
8.451 8.455
-492
-496
631
-535
-570
-574
-607
,611
8-459
400
410
420
430
440
8.644
450
460
470
480
490
8.814
-845
47'7
a907
-937
500
510
520
530
540
8.966 8.969 8.972 8.974
-994
-997 9.000 9.003
9.022 9.025
-031
,028
,050
*053
-058
so55
,077
-079
-082
-085
7.238 7.248 7.257
-331
-349
,340
-418
-409
-435
-426
-492
-500
-516
-508
~570 4577
492
,585
a679
4'14
,748
*781
7-972 7.977 7.983 7.989 7,994
8.028 8.033 8.039 8-044 8.050
,082
-087
-103
-093
-098
~140 ~145 -150
,155
-134
-190
-185
-195 -200
,205
8.243 8.248
-290 *295
-335
-340
-384
-379
-422
*426
8.253
,299
0344
-388
430
8.000
-055
-109
el60
-209
8-257 8-262 8.267
-304
,313
*308
a349
-358
-353
-392
,401
-397
-435 -439
-443
8.271
,317
-862
405
-47
8-464 8-468 8.472 8.476 8-480 8.484 8.488
-516
-504 -508
-520
0512
-524
-527
-539
-543
-558
~551 -555
,562
-547
0566
-577
-592 a596
681
-600
-585
0589
4304
-615
,629
a618
-633
-622 *626
-637
-640
600
8.647 8.651 8.655 8.658 8.662
-683
-687 -690
*697
-693
-718
-725
-721
-731
-728
*752
-758
.755
-762
4'65
~785 4'88
-791
4'98
-794
8.665
8.817
8.833
-864
4495
8.820 8.823
0852
,855
-883
-886
-910
-913
-916
-940
-943 -945
-849
480
8.827
-858
-889
-919
448
8.830
-861
0892
-922
951
8-669 8.672
-707
-704
-741
-735
4'38
-768
,775
-771
-801
-804
407
-700
8.836
467
-898
-925
428
-957
-954
8.977 8-980 8.983
9.006 9.008 9.011
-033
a039
,036
-061
-066
-063
-093
-087
a090
8.676
-711
-745
-778
0811
-870
8.842
-873
.go1
-931
-960
434
463
8.839
0904
8.986 8-989 8.992
9.014 9.017 9.020
-042
,047
-044
-069
-074
*Or1
*lo1
a085
,098
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Logarithms to base 2
225
Logarithms to base 2
8
0
1
2
550
560
570
580
590
9-103
-129
-155
-180
-205
9.106
-132
.157
-182
,207
9.109
-134
-160
-185
-210
4
3
9-111 9.114
el40
-137
~162 -165
~ 1 8 7 -190
*214
-212
9.116
-142
el67
-192
-217
9.119
9.127
9.122 9.124
-152
~145 ,147
-150
-170
-175
~177
-172
-202
-195
~ 1 9 7 -200
-222
-219
~ 2 2 4 -226
600
610
620
630
640
9.229
-253
-276
-299
-322
9.231
-255
-278
-301
,324
9-234
-257
-281
-304
-326
9-236
-260
-283
-306
-329
9.238
-262
-285
,308
-331
9.241
-264
-288
-311
-333
9.243
-267
-290
-313
*335
9.246
-269
-292
*315
,338
9.248
-271
-295
-317
-340
9.250
-274
-297
-320
*342
650
660
670
680
690
9-344l
-366
-388
-409
-430
9.a7
9.349
-371
~369
-392
-390
*412
-414
433
-435
9.351
-373
-394
-416
-437
9.353
.375
-397
-418
-439
9.355
-377
*399
-420
441
9-358
-379
-401
-422
-4443
9.360
0382
0403
a424
,445
9.362
-384
-405
-426
-447
9.36.4
-386
-407
-428
449
700
710
720
730
740
9-451
-472
*492
-512
-531
9.453
-474
-494
-514
-533
9.455
-476
-496
*516
*535
9.457
-478
-498
518
-537
9.459
*480
-500
-520
-539
9.461 9-464
-484
-482
-504
-502
~522
-524
-541
-543
9.466
*486
-506
-526
-545
9.468
-488
*508
-527
-547
9.470
-490
-510
-529
-549
750
760
770
780
790
9.551
-570
-589
-607
-626
9.553
-572
0591
,609
,628
9.555
-574
-592
-611
-629
9.557
-576
-594
-613
-631
9.558
-577
,596
-615
6633
9.560
-579
*598
-617
-635
9.564
0583
-602
-620
*638
9.566
-585
-604
-622
-640
9-568
-587
a605
-624
-642
800
810
820
830
840
9.644
-662
-679
-697
-714
9.646
9.647
-665
a664
-681
-683
~699
-700
-716
-718
9.649
-667
-685
TO2
*719
9.651 9.653
-671
*669
-688
a687
4'06
-704
*723
~721
9-655 9,656
*674
-672
~690 -692
a709
-707
-725
4'26
9.658
-676
-693
,711
a728
9.660
-678
-695
-713
-730
850
860
870
880
890
9.731
-748
-765
-781
-798
9-733
*750
4'67
-783
a799
9.735
-752
-768
9.738
401
9-736
-753
-770
-786
-803
m771
-788
-804
9-740
-757
-773
-790
-806
9.741
-758
*775
-791
-807
9.743
4'60
4"76
-793
-809
9.745
-762
*778
-794
411
9.747
-763
-780
-796
-812
900
910
920
930
940
9.814
-830
-845
-861
-877
9.815
-831
,847
463
0878
9.817
433
-849
-864
-880
9.819
-834
-850
-866
-881
9.820
436
a852
,867
-883
9.822
-838
-853
-869
-884
9.823
-839
-855
-870
-886
9.825
4341
-856
-872
487
9.827
-842
458
-873
-889
9.828
950
960
970
980
990
9.892
0907
-922
-937
-951
9.893
a908
-923
,938
,953
9-895
-910
-925
-940
-954
9.896
*911
-926
-941
-956
9,898
-913
-928
-943
-957
9.899
-914
-929
-944
.959
9.901
-916
-931
-945
-960
9.902
-917
-932
-947
-961
9.904
-919
,934
-948
-963
9.905
a920
*935
-950
a964
108
104
106
lo"
109
1010
10
3-322
10%
6.644
-785
-755
5
6
9.562
-581
-600
-618
-637
107
7
108
9.966 13.288 16.610 19.932 23.253 26.575
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9
-8U
460
-875
,890
29.897 33.219