Comprehensive Log₁₀ Calculator Guide Using Prime Numbers (1000-Prime Table)

A Brief History of Logarithms

How logarithms transformed calculation — from Napier’s tables to modern applications.

Logarithms (base 10) — visual banner for the post

The idea of logarithms was introduced to simplify complex calculations by converting multiplication into addition and division into subtraction. The most influential originator was John Napier, a Scottish mathematician who published Mirifici Logarithmorum Canonis Descriptio in 1614. Napier’s work provided tables that dramatically reduced the time required for astronomical and navigational computations.

Soon after Napier, Henry Briggs collaborated to refine the system into common logarithms (base 10), which became the standard for scientists and engineers. Briggs published extensive logarithm tables and promoted the use of base-10 logs because they matched the decimal number system and made hand calculations straightforward.

For centuries, log tables and the slide rule were essential tools for astronomy, navigation, engineering, and applied mathematics. Before electronic calculators, students and professionals relied on these tables to compute powers, roots, and products quickly and accurately.

In the 20th century, the rise of electronic calculators and computers changed how we compute, but the mathematical importance of logarithms remained. Today, logarithms underpin many modern fields — from numerical methods and algorithm analysis to data science, signal processing, and cryptography. The same properties Napier exploited (the product and power rules) are foundational in contemporary algorithms and number theory.

Key takeaway: Napier’s insight into transforming multiplicative problems into additive ones created a computational revolution. Thanks to the work of Napier and Briggs, logarithm tables and the common log (log₁₀) became standard tools — and their legacy lives on in modern mathematics and computing.

How to find log₁₀ of any number using prime factorization

Use the product rule of logarithms and the fact every number is built from prime factors.

Key idea

The central property we use is the logarithm product rule:
log₁₀(x·y) = log₁₀(x) + log₁₀(y).

Every positive integer can be written uniquely as a product of primes:
prime factorization — for example \( 360 = 2^3 \cdot 3^2 \cdot 5 \). Using the product rule repeatedly gives
\(\displaystyle \log_{10}(360)=\log_{10}(2^3)+\log_{10}(3^2)\)
\(+\log_{10}(5)\).

Important: Use the power rule of logs to handle exponents:

log₁₀(a^k) = k · log₁₀(a)

Why this is useful

1. find its prime factorization \(n = p_1^{a_1} p_2^{a_2}\cdots p_k^{a_k}\),
2. use \(\log_{10}(n) = a_1·\log_{10}(p_1) \)
   \(+ a_2·\log_{10}(p_2) + \dots + a_k·\log_{10}(p_k)\).

This avoids long decimal log table lookups or calculators when you already have prime logs stored.

Two worked examples

Example 1 — \( \log_{10}(360) \)

Prime factorize: \(360 = 2^3 \cdot 3^2 \cdot 5\). Using the power rule and product rule:

\(\log_{10}(360) = 3\log_{10}(2) + 2\log_{10}(3) \)
\( + \log_{10}(5)\)

Substitute known values (rounded to 5 decimals):

• \(\log_{10}(2) = 0.30103\)
• \(\log_{10}(3) = 0.47712\)
• \(\log_{10}(5) = 0.69897\)

Calculation:

\(3\times0.30103 = 0.90309\)
\(2\times0.47712 = 0.95424\)
\(1\times0.69897 = 0.69897\)

Sum: \(0.90309 + 0.95424 + 0.69897 \)
\(= 2.55630\).

So \(\log_{10}(360) \approx 2.55630\), and \(10^{2.55630}\approx360\).

Example 2 — \( \log_{10}(2023) \)

Factorize: \(2023 = 7 \cdot 17^2\) (since \(17^2=289\) and \(289\cdot7=2023\)). Apply the rules:

\(\log_{10}(2023) = \log_{10}(7) + 2\log_{10}(17)\)

Substitute known values (rounded to 5 decimals):

• \(\log_{10}(7) = 0.84510\)
• \(\log_{10}(17) = 1.23045\)

\( \log_{10}(2023) = 0.84510 + 2\times1.23045 = 0.84510 + 2.46090 = 3.30600 \).

So \(\log_{10}(2023) \approx 3.30600\), and \(10^{3.30600}\approx2023\).

Extra notes & tips

• Zeros at the end: If a number has trailing zeros, each factor of 10 contributes exactly +1 to the base-10 logarithm. Example: \( \log_{10}(2500) = \log_{10}(25) + \log_{10}(100) = 2\log_{10}(5) + 2\).
• Accuracy: Keep the prime logs to enough decimal places for your required precision. We used 5 decimal places above; use more if you need higher accuracy.
• Efficiency: Store a small lookup table of \(\log_{10}\) for primes (first 1000 primes is excellent) — then any composite number’s log is a few table lookups plus some multiplications and additions.

Try it yourself: pick any integer, factor it into primes, and add the logs of those primes multiplied by their exponents.

— Learn4Math

S.No.	Prime	log₁₀(p)
1	2	0.30103
2	3	0.47712
3	5	0.69897
4	7	0.84510
5	11	1.04139
6	13	1.11394
7	17	1.23045
8	19	1.27875
9	23	1.36173
10	29	1.46240
11	31	1.49136
12	37	1.56820
13	41	1.61278
14	43	1.63347
15	47	1.67210
16	53	1.72428
17	59	1.77085
18	61	1.78533
19	67	1.82607
20	71	1.85126
21	73	1.86332
22	79	1.89763
23	83	1.91908
24	89	1.94939
25	97	1.98677
26	101	2.00432
27	103	2.01284
28	107	2.02938
29	109	2.03743
30	113	2.05308
31	127	2.10380
32	131	2.11727
33	137	2.13672
34	139	2.14301
35	149	2.17319
36	151	2.17898
37	157	2.19590
38	163	2.21219
39	167	2.22272
40	173	2.23805
41	179	2.25285
42	181	2.25768
43	191	2.28103
44	193	2.28556
45	197	2.29447
46	199	2.29885
47	211	2.32428
48	223	2.34830
49	227	2.35603
50	229	2.35984
51	233	2.36736
52	239	2.37840
53	241	2.38202
54	251	2.39967
55	257	2.40993
56	263	2.41996
57	269	2.42975
58	271	2.43297
59	277	2.44248
60	281	2.44871
61	283	2.45179
62	293	2.46687
63	307	2.48714
64	311	2.49276
65	313	2.49554
66	317	2.50106
67	331	2.51983
68	337	2.52763
69	347	2.54033
70	349	2.54283
71	353	2.54777
72	359	2.55509
73	367	2.56467
74	373	2.57171
75	379	2.57864
76	383	2.58320
77	389	2.58995
78	397	2.59879
79	401	2.60314
80	409	2.61172
81	419	2.62221
82	421	2.62428
83	431	2.63448
84	433	2.63649
85	439	2.64246
86	443	2.64640
87	449	2.65225
88	457	2.65992
89	461	2.66370
90	463	2.66558
91	467	2.66932
92	479	2.68034
93	487	2.68753
94	491	2.69108
95	499	2.69810
96	503	2.70157
97	509	2.70672
98	521	2.71684
99	523	2.71850
100	541	2.73320
101	547	2.73799
102	557	2.74586
103	563	2.75051
104	569	2.75511
105	571	2.75664
106	577	2.76118
107	587	2.76864
108	593	2.77305
109	599	2.77743
110	601	2.77887
111	607	2.78319
112	613	2.78746
113	617	2.79029
114	619	2.79169
115	631	2.80003
116	641	2.80686
117	643	2.80821
118	647	2.81090
119	653	2.81491
120	659	2.81889
121	661	2.82020
122	673	2.82802
123	677	2.83059
124	683	2.83442
125	691	2.83948
126	701	2.84572
127	709	2.85065
128	719	2.85673
129	727	2.86153
130	733	2.86510
131	739	2.86864
132	743	2.87099
133	751	2.87564
134	757	2.87910
135	761	2.88138
136	769	2.88593
137	773	2.88818
138	787	2.89597
139	797	2.90146
140	809	2.90795
141	811	2.90902
142	821	2.91434
143	823	2.91540
144	827	2.91751
145	829	2.91855
146	839	2.92376
147	853	2.93095
148	857	2.93298
149	859	2.93399
150	863	2.93601
151	877	2.94300
152	881	2.94498
153	883	2.94596
154	887	2.94792
155	907	2.95761
156	911	2.95952
157	919	2.96332
158	929	2.96802
159	937	2.97174
160	941	2.97359
161	947	2.97635
162	953	2.97909
163	967	2.98543
164	971	2.98722
165	977	2.98989
166	983	2.99255
167	991	2.99607
168	997	2.99870
169	1009	3.00389
170	1013	3.00561
171	1019	3.00817
172	1021	3.00903
173	1031	3.01326
174	1033	3.01410
175	1039	3.01662
176	1049	3.02078
177	1051	3.02160
178	1061	3.02572
179	1063	3.02653
180	1069	3.02898
181	1087	3.03623
182	1091	3.03782
183	1093	3.03862
184	1097	3.04021
185	1103	3.04258
186	1109	3.04493
187	1117	3.04805
188	1123	3.05038
189	1129	3.05269
190	1151	3.06108
191	1153	3.06183
192	1163	3.06558
193	1171	3.06856
194	1181	3.07225
195	1187	3.07445
196	1193	3.07664
197	1201	3.07954
198	1213	3.08386
199	1217	3.08529
200	1223	3.08743
201	1229	3.08955
202	1231	3.09026
203	1237	3.09237
204	1249	3.09656
205	1259	3.10003
206	1277	3.10619
207	1279	3.10687
208	1283	3.10823
209	1289	3.11025
210	1291	3.11093
211	1297	3.11294
212	1301	3.11428
213	1303	3.11494
214	1307	3.11628
215	1319	3.12024
216	1321	3.12090
217	1327	3.12287
218	1361	3.13386
219	1367	3.13577
220	1373	3.13767
221	1381	3.14019
222	1399	3.14582
223	1409	3.14891
224	1423	3.15320
225	1427	3.15442
226	1429	3.15503
227	1433	3.15625
228	1439	3.15806
229	1447	3.16047
230	1451	3.16167
231	1453	3.16227
232	1459	3.16406
233	1471	3.16761
234	1481	3.17056
235	1483	3.17114
236	1487	3.17231
237	1489	3.17289
238	1493	3.17406
239	1499	3.17580
240	1511	3.17926
241	1523	3.18270
242	1531	3.18498
243	1543	3.18837
244	1549	3.19005
245	1553	3.19117
246	1559	3.19285
247	1567	3.19507
248	1571	3.19618
249	1579	3.19838
250	1583	3.19948
251	1597	3.20330
252	1601	3.20439
253	1607	3.20602
254	1609	3.20656
255	1613	3.20763
256	1619	3.20925
257	1621	3.20978
258	1627	3.21139
259	1637	3.21405
260	1657	3.21932
261	1663	3.22089
262	1667	3.22194
263	1669	3.22246
264	1693	3.22866
265	1697	3.22968
266	1699	3.23019
267	1709	3.23274
268	1721	3.23578
269	1723	3.23629
270	1733	3.23880
271	1741	3.24080
272	1747	3.24229
273	1753	3.24378
274	1759	3.24527
275	1777	3.24969
276	1783	3.25115
277	1787	3.25212
278	1789	3.25261
279	1801	3.25551
280	1811	3.25792
281	1823	3.26079
282	1831	3.26269
283	1847	3.26647
284	1861	3.26975
285	1867	3.27114
286	1871	3.27207
287	1873	3.27254
288	1877	3.27346
289	1879	3.27393
290	1889	3.27623
291	1901	3.27898
292	1907	3.28035
293	1913	3.28171
294	1931	3.28578
295	1933	3.28623
296	1949	3.28981
297	1951	3.29026
298	1973	3.29513
299	1979	3.29645
300	1987	3.29820
301	1993	3.29951
302	1997	3.30038
303	1999	3.30081
304	2003	3.30168
305	2011	3.30341
306	2017	3.30471
307	2027	3.30685
308	2029	3.30728
309	2039	3.30942
310	2053	3.31239
311	2063	3.31450
312	2069	3.31576
313	2081	3.31827
314	2083	3.31869
315	2087	3.31952
316	2089	3.31994
317	2099	3.32201
318	2111	3.32449
319	2113	3.32490
320	2129	3.32818
321	2131	3.32858
322	2137	3.32980
323	2141	3.33062
324	2143	3.33102
325	2153	3.33304
326	2161	3.33465
327	2179	3.33826
328	2203	3.34301
329	2207	3.34380
330	2213	3.34498
331	2221	3.34655
332	2237	3.34967
333	2239	3.35005
334	2243	3.35083
335	2251	3.35238
336	2267	3.35545
337	2269	3.35583
338	2273	3.35660
339	2281	3.35813
340	2287	3.35927
341	2293	3.36040
342	2297	3.36116
343	2309	3.36342
344	2311	3.36380
345	2333	3.36791
346	2339	3.36903
347	2341	3.36940
348	2347	3.37051
349	2351	3.37125
350	2357	3.37236
351	2371	3.37493
352	2377	3.37603
353	2381	3.37676
354	2383	3.37712
355	2389	3.37822
356	2393	3.37894
357	2399	3.38003
358	2411	3.38220
359	2417	3.38328
360	2423	3.38435
361	2437	3.38686
362	2441	3.38757
363	2447	3.38863
364	2459	3.39076
365	2467	3.39217
366	2473	3.39322
367	2477	3.39393
368	2503	3.39846
369	2521	3.40157
370	2531	3.40329
371	2539	3.40466
372	2543	3.40535
373	2549	3.40637
374	2551	3.40671
375	2557	3.40773
376	2579	3.41145
377	2591	3.41347
378	2593	3.41380
379	2609	3.41647
380	2617	3.41780
381	2621	3.41847
382	2633	3.42045
383	2647	3.42275
384	2657	3.42439
385	2659	3.42472
386	2663	3.42537
387	2671	3.42667
388	2677	3.42765
389	2683	3.42862
390	2687	3.42927
391	2689	3.42959
392	2693	3.43024
393	2699	3.43120
394	2707	3.43249
395	2711	3.43313
396	2713	3.43345
397	2719	3.43441
398	2729	3.43600
399	2731	3.43632
400	2741	3.43791
401	2749	3.43917
402	2753	3.43981
403	2767	3.44201
404	2777	3.44358
405	2789	3.44545
406	2791	3.44576
407	2797	3.44669
408	2801	3.44731
409	2803	3.44762
410	2819	3.45010
411	2833	3.45225
412	2837	3.45286
413	2843	3.45378
414	2851	3.45500
415	2857	3.45591
416	2861	3.45652
417	2879	3.45924
418	2887	3.46045
419	2897	3.46195
420	2903	3.46285
421	2909	3.46374
422	2917	3.46494
423	2927	3.46642
424	2939	3.46820
425	2953	3.47026
426	2957	3.47085
427	2963	3.47173
428	2969	3.47261
429	2971	3.47290
430	2999	3.47698
431	3001	3.47727
432	3011	3.47871
433	3019	3.47986
434	3023	3.48044
435	3037	3.48244
436	3041	3.48302
437	3049	3.48416
438	3061	3.48586
439	3067	3.48671
440	3079	3.48841
441	3083	3.48897
442	3089	3.48982
443	3109	3.49262
444	3119	3.49402
445	3121	3.49429
446	3137	3.49651
447	3163	3.50010
448	3167	3.50065
449	3169	3.50092
450	3181	3.50256
451	3187	3.50338
452	3191	3.50393
453	3203	3.50556
454	3209	3.50637
455	3217	3.50745
456	3221	3.50799
457	3229	3.50907
458	3251	3.51202
459	3253	3.51228
460	3257	3.51282
461	3259	3.51308
462	3271	3.51468
463	3299	3.51838
464	3301	3.51865
465	3307	3.51943
466	3313	3.52022
467	3319	3.52101
468	3323	3.52153
469	3329	3.52231
470	3331	3.52257
471	3343	3.52414
472	3347	3.52466
473	3359	3.52621
474	3361	3.52647
475	3371	3.52776
476	3373	3.52802
477	3389	3.53007
478	3391	3.53033
479	3407	3.53237
480	3413	3.53314
481	3433	3.53567
482	3449	3.53769
483	3457	3.53870
484	3461	3.53920
485	3463	3.53945
486	3467	3.53995
487	3469	3.54020
488	3491	3.54295
489	3499	3.54394
490	3511	3.54543
491	3517	3.54617
492	3527	3.54741
493	3529	3.54765
494	3533	3.54814
495	3539	3.54888
496	3541	3.54913
497	3547	3.54986
498	3557	3.55108
499	3559	3.55133
500	3571	3.55279
501	3581	3.55400
502	3583	3.55425
503	3593	3.55546
504	3607	3.55715
505	3613	3.55787
506	3617	3.55835
507	3623	3.55907
508	3631	3.56003
509	3637	3.56074
510	3643	3.56146
511	3659	3.56336
512	3671	3.56478
513	3673	3.56502
514	3677	3.56549
515	3691	3.56714
516	3697	3.56785
517	3701	3.56832
518	3709	3.56926
519	3719	3.57043
520	3727	3.57136
521	3733	3.57206
522	3739	3.57276
523	3761	3.57530
524	3767	3.57600
525	3769	3.57623
526	3779	3.57738
527	3793	3.57898
528	3797	3.57944
529	3803	3.58013
530	3821	3.58218
531	3823	3.58240
532	3833	3.58354
533	3847	3.58512
534	3851	3.58557
535	3853	3.58580
536	3863	3.58692
537	3877	3.58850
538	3881	3.58894
539	3889	3.58984
540	3907	3.59184
541	3911	3.59229
542	3917	3.59295
543	3919	3.59318
544	3923	3.59362
545	3929	3.59428
546	3931	3.59450
547	3943	3.59583
548	3947	3.59627
549	3967	3.59846
550	3989	3.60086
551	4001	3.60217
552	4003	3.60239
553	4007	3.60282
554	4013	3.60347
555	4019	3.60412
556	4021	3.60433
557	4027	3.60498
558	4049	3.60735
559	4051	3.60756
560	4057	3.60821
561	4073	3.60991
562	4079	3.61055
563	4091	3.61183
564	4093	3.61204
565	4099	3.61268
566	4111	3.61395
567	4127	3.61563
568	4129	3.61584
569	4133	3.61627
570	4139	3.61690
571	4153	3.61836
572	4157	3.61878
573	4159	3.61899
574	4177	3.62086
575	4201	3.62335
576	4211	3.62439
577	4217	3.62500
578	4219	3.62521
579	4229	3.62624
580	4231	3.62644
581	4241	3.62747
582	4243	3.62767
583	4253	3.62870
584	4259	3.62931
585	4261	3.62951
586	4271	3.63053
587	4273	3.63073
588	4283	3.63175
589	4289	3.63236
590	4297	3.63317
591	4327	3.63619
592	4337	3.63719
593	4339	3.63739
594	4349	3.63839
595	4357	3.63919
596	4363	3.63979
597	4373	3.64078
598	4391	3.64256
599	4397	3.64316
600	4409	3.64434
601	4421	3.64552
602	4423	3.64572
603	4441	3.64748
604	4447	3.64807
605	4451	3.64846
606	4457	3.64904
607	4463	3.64963
608	4481	3.65137
609	4483	3.65157
610	4493	3.65254
611	4507	3.65389
612	4513	3.65447
613	4517	3.65485
614	4519	3.65504
615	4523	3.65543
616	4547	3.65772
617	4549	3.65792
618	4561	3.65906
619	4567	3.65963
620	4583	3.66115
621	4591	3.66191
622	4597	3.66247
623	4603	3.66304
624	4621	3.66474
625	4637	3.66624
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