x	log₁₀x or log x										Mean Differences (Add)
x	0	1	2	3	4	5	6	7	8	9	1	2	3	4	5	6	7	8	9	10
1.0	0.0000	0.0043	0.0086	0.0128	0.0170	0.0212	0.0253	0.0294	0.0334	0.0374	4	9	13	17	21	26	30	34	38	43
1.1	0.0414	0.0453	0.0492	0.0531	0.0569	0.0607	0.0645	0.0682	0.0719	0.0755	4	8	12	16	20	24	28	32	36	40
1.2	0.0792	0.0828	0.0864	0.0899	0.0934	0.0969	0.1004	0.1038	0.1072	0.1106	3	7	11	14	18	22	25	29	33	36
1.3	0.1139	0.1173	0.1206	0.1239	0.1271	0.1303	0.1335	0.1367	0.1399	0.1430	3	6	9	13	16	19	22	25	28	31
1.4	0.1461	0.1492	0.1523	0.1553	0.1584	0.1614	0.1644	0.1673	0.1703	0.1732	3	6	9	12	15	18	21	24	27	30
1.5	0.1761	0.1790	0.1818	0.1847	0.1875	0.1903	0.1931	0.1959	0.1987	0.2014	3	6	9	11	14	17	20	23	26	29
1.6	0.2041	0.2068	0.2095	0.2122	0.2148	0.2175	0.2201	0.2227	0.2253	0.2279	3	5	8	11	14	17	20	22	25	28
1.7	0.2304	0.2330	0.2355	0.2380	0.2405	0.2430	0.2455	0.2480	0.2504	0.2529	3	5	8	11	14	16	19	22	24	27
1.8	0.2553	0.2577	0.2601	0.2625	0.2648	0.2672	0.2695	0.2718	0.2742	0.2765	2	5	7	10	13	15	18	20	23	25
1.9	0.2788	0.2810	0.2833	0.2856	0.2878	0.2900	0.2923	0.2945	0.2967	0.3010	2	5	7	9	12	14	16	19	21	23
2.0	0.3010	0.3032	0.3054	0.3075	0.3096	0.3118	0.3139	0.3160	0.3181	0.3201	2	4	6	9	11	13	15	17	19	21
2.1	0.3222	0.3243	0.3263	0.3284	0.3304	0.3324	0.3345	0.3365	0.3385	0.3404	2	4	6	8	10	12	14	16	18	20
2.2	0.3424	0.3444	0.3464	0.3483	0.3502	0.3522	0.3541	0.3560	0.3579	0.3598	2	4	6	8	10	12	14	15	17	19
2.3	0.3617	0.3636	0.3655	0.3674	0.3692	0.3711	0.3729	0.3747	0.3766	0.3784	2	4	6	7	9	11	13	15	17	19
2.4	0.3802	0.3820	0.3838	0.3856	0.3874	0.3892	0.3910	0.3927	0.3945	0.3962	2	4	5	7	9	11	13	14	16	18
2.5	0.3979	0.3997	0.4014	0.4031	0.4048	0.4065	0.4082	0.4099	0.4116	0.4133	2	3	5	7	9	10	12	14	16	17
2.6	0.4150	0.4166	0.4183	0.4200	0.4216	0.4232	0.4249	0.4265	0.4281	0.4298	2	3	5	7	8	10	12	13	15	17
2.7	0.4314	0.4330	0.4346	0.4362	0.4378	0.4393	0.4409	0.4425	0.4440	0.4456	1	3	5	6	8	10	11	13	15	16
2.8	0.4472	0.4487	0.4502	0.4518	0.4533	0.4548	0.4564	0.4579	0.4594	0.4609	1	3	4	6	8	9	11	13	14	16
2.9	0.4624	0.4639	0.4654	0.4669	0.4683	0.4698	0.4713	0.4728	0.4742	0.4757	1	3	4	6	7	9	10	12	13	15
3.0	0.4771	0.4786	0.4800	0.4814	0.4829	0.4843	0.4857	0.4871	0.4885	0.4900	1	3	4	5	7	8	10	11	13	14
3.1	0.4914	0.4928	0.4942	0.4955	0.4969	0.4983	0.4997	0.5011	0.5024	0.5038	1	2	4	5	7	8	9	11	12	14
3.2	0.5051	0.5065	0.5079	0.5092	0.5105	0.5119	0.5132	0.5145	0.5159	0.5172	1	2	4	5	6	8	9	11	12	13
3.3	0.5185	0.5198	0.5211	0.5224	0.5237	0.5250	0.5263	0.5276	0.5289	0.5302	1	2	3	5	6	7	9	10	11	13
3.4	0.5315	0.5328	0.5340	0.5353	0.5366	0.5378	0.5391	0.5403	0.5416	0.5428	1	2	3	4	6	7	8	9	11	12
3.5	0.5441	0.5453	0.5465	0.5478	0.5490	0.5502	0.5514	0.5527	0.5539	0.5551	1	2	3	4	5	7	8	9	10	11
3.6	0.5563	0.5575	0.5587	0.5599	0.5611	0.5623	0.5635	0.5647	0.5658	0.5670	1	2	3	4	5	6	7	8	9	10
3.7	0.5682	0.5694	0.5705	0.5717	0.5729	0.5740	0.5752	0.5763	0.5775	0.5786	1	2	3	4	5	6	7	8	9	10
3.8	0.5798	0.5809	0.5821	0.5832	0.5843	0.5855	0.5866	0.5877	0.5888	0.5900	1	2	3	4	5	6	7	8	9	10
3.9	0.5911	0.5922	0.5933	0.5944	0.5955	0.5966	0.5977	0.5988	0.5999	0.6010	1	2	3	4	5	6	7	8	9	10

Mathematical Tables - Logarithms (Page 2)

Mathematical Tables
Common Logarithms of Numbers (Page 2)

x	log₁₀x or log x										Mean Differences (Add)
x	0	1	2	3	4	5	6	7	8	9	1	2	3	4	5	6	7	8	9	10
4.0	0.6021	0.6031	0.6042	0.6053	0.6064	0.6075	0.6085	0.6095	0.6106	0.6117	1	2	3	4	5	6	7	8	9	9
4.1	0.6128	0.6138	0.6149	0.6159	0.6170	0.6180	0.6191	0.6201	0.6212	0.6222	1	2	3	4	5	6	7	8	9	9
4.2	0.6232	0.6243	0.6253	0.6263	0.6274	0.6284	0.6294	0.6304	0.6314	0.6325	1	2	3	4	5	6	7	8	9	9
4.3	0.6335	0.6345	0.6355	0.6365	0.6375	0.6385	0.6395	0.6405	0.6415	0.6425	1	2	3	4	5	6	7	8	9	9
4.4	0.6435	0.6444	0.6454	0.6464	0.6474	0.6484	0.6494	0.6503	0.6513	0.6523	1	2	3	4	5	6	7	8	9	9
4.5	0.6532	0.6542	0.6552	0.6561	0.6571	0.6580	0.6590	0.6599	0.6609	0.6618	1	2	3	4	5	6	7	8	9	9
4.6	0.6628	0.6637	0.6646	0.6656	0.6665	0.6674	0.6684	0.6693	0.6702	0.6712	1	2	3	4	5	6	7	8	9	9
4.7	0.6721	0.6730	0.6740	0.6749	0.6758	0.6767	0.6776	0.6786	0.6795	0.6804	1	2	3	4	5	6	7	8	9	9
4.8	0.6812	0.6821	0.6830	0.6839	0.6848	0.6856	0.6865	0.6874	0.6883	0.6892	1	2	3	4	5	6	7	8	9	9
4.9	0.6902	0.6910	0.6919	0.6928	0.6937	0.6946	0.6955	0.6964	0.6972	0.6981	1	2	3	4	5	6	7	8	9	9
5.0	0.6990	0.6998	0.7006	0.7016	0.7023	0.7031	0.7038	0.7046	0.7054	0.7062	1	2	3	4	5	6	7	8	9	9
5.1	0.7070	0.7078	0.7086	0.7093	0.7100	0.7108	0.7115	0.7123	0.7130	0.7138	1	2	3	4	5	6	7	8	9	9
5.2	0.7145	0.7152	0.7159	0.7166	0.7173	0.7180	0.7187	0.7194	0.7201	0.7208	1	2	3	4	5	6	7	8	9	9
5.3	0.7215	0.7222	0.7229	0.7236	0.7243	0.7250	0.7256	0.7263	0.7270	0.7277	1	2	3	4	5	6	7	8	9	9
5.4	0.7284	0.7291	0.7298	0.7304	0.7311	0.7318	0.7324	0.7331	0.7338	0.7344	1	2	3	4	5	6	7	8	9	9
5.5	0.7351	0.7357	0.7364	0.7370	0.7377	0.7383	0.7390	0.7396	0.7403	0.7409	1	2	3	4	5	6	7	8	9	9
5.6	0.7415	0.7421	0.7428	0.7434	0.7440	0.7446	0.7452	0.7458	0.7464	0.7470	1	2	3	4	5	6	7	8	9	9
5.7	0.7475	0.7481	0.7487	0.7493	0.7499	0.7505	0.7511	0.7517	0.7523	0.7528	1	2	3	4	5	6	7	8	9	9
5.8	0.7534	0.7540	0.7545	0.7551	0.7557	0.7563	0.7568	0.7574	0.7580	0.7585	1	2	3	4	5	6	7	8	9	9
5.9	0.7591	0.7596	0.7602	0.7608	0.7614	0.7619	0.7625	0.7630	0.7636	0.7642	1	2	3	4	5	6	7	8	9	9
6.0	0.7647	0.7653	0.7658	0.7664	0.7669	0.7675	0.7680	0.7686	0.7691	0.7697	1	2	3	4	5	6	7	8	9	9
6.1	0.7702	0.7708	0.7713	0.7719	0.7724	0.7730	0.7735	0.7741	0.7746	0.7752	1	2	3	4	5	6	7	8	9	10
6.2	0.7924	0.7929	0.7935	0.7940	0.7945	0.7951	0.7956	0.7961	0.7966	0.7972	1	2	3	4	5	6	7	8	9	10
6.3	0.7993	0.7998	0.8003	0.8008	0.8013	0.8018	0.8023	0.8028	0.8033	0.8038	1	2	3	4	5	6	7	8	9	10
6.4	0.8062	0.8067	0.8072	0.8077	0.8082	0.8087	0.8092	0.8097	0.8102	0.8107	1	2	3	4	5	6	7	8	9	10
6.5	0.8122	0.8127	0.8132	0.8137	0.8142	0.8147	0.8152	0.8157	0.8162	0.8166	1	2	3	4	5	6	7	8	9	10
6.6	0.8195	0.8200	0.8204	0.8209	0.8214	0.8219	0.8223	0.8228	0.8233	0.8237	1	2	3	4	5	6	7	8	9	10
6.7	0.8261	0.8265	0.8270	0.8274	0.8279	0.8283	0.8288	0.8292	0.8297	0.8301	1	2	3	4	5	6	7	8	9	10
6.8	0.8325	0.8329	0.8333	0.8337	0.8342	0.8346	0.8350	0.8354	0.8358	0.8362	1	2	3	4	5	6	7	8	9	10
6.9	0.8388	0.8392	0.8396	0.8400	0.8404	0.8408	0.8412	0.8416	0.8420	0.8424	1	2	3	4	5	6	7	8	9	10
7.0	0.8451	0.8455	0.8459	0.8463	0.8467	0.8471	0.8475	0.8479	0.8483	0.8487	1	2	3	4	5	6	7	8	9	10
7.1	0.8513	0.8517	0.8520	0.8524	0.8528	0.8532	0.8536	0.8540	0.8543	0.8547	1	2	3	4	5	6	7	8	9	10
7.2	0.8573	0.8576	0.8580	0.8584	0.8588	0.8591	0.8595	0.8599	0.8603	0.8606	1	2	3	4	5	6	7	8	9	10
7.3	0.8633	0.8637	0.8640	0.8644	0.8648	0.8651	0.8655	0.8659	0.8662	0.8666	1	2	3	4	5	6	7	8	9	10
7.4	0.8692	0.8696	0.8699	0.8703	0.8706	0.8710	0.8713	0.8717	0.8720	0.8724	1	2	3	4	5	6	7	8	9	10
7.5	0.8751	0.8754	0.8757	0.8761	0.8764	0.8767	0.8771	0.8774	0.8778	0.8781	1	2	3	4	5	6	7	8	9	10
7.6	0.8802	0.8805	0.8809	0.8812	0.8815	0.8819	0.8822	0.8825	0.8829	0.8832	1	2	3	4	5	6	7	8	9	10
7.7	0.8865	0.8868	0.8871	0.8875	0.8878	0.8881	0.8884	0.8888	0.8891	0.8894	1	2	3	4	5	6	7	8	9	10
7.8	0.8921	0.8924	0.8927	0.8930	0.8933	0.8936	0.8939	0.8942	0.8945	0.8948	1	2	3	4	5	6	7	8	9	10
7.9	0.8976	0.8979	0.8982	0.8985	0.8988	0.8991	0.8994	0.8997	0.9000	0.9003	1	2	3	4	5	6	7	8	9	10

Mathematical Tables - Logarithms (Page 3)

Mathematical Tables
Common Logarithms of Numbers (Page 3)

x	log₁₀x or log x										Mean Differences (Add)
x	0	1	2	3	4	5	6	7	8	9	1	2	3	4	5	6	7	8	9	10
8.0	0.9031	0.9036	0.9042	0.9047	0.9053	0.9058	0.9063	0.9069	0.9074	0.9079	1	2	3	3	4	4	5	5	6	7
8.1	0.9085	0.9090	0.9095	0.9101	0.9106	0.9112	0.9117	0.9123	0.9128	0.9133	1	2	2	3	3	4	4	5	5	6
8.2	0.9138	0.9143	0.9149	0.9154	0.9160	0.9165	0.9170	0.9176	0.9181	0.9186	1	2	2	3	3	4	4	5	5	6
8.3	0.9191	0.9196	0.9201	0.9206	0.9212	0.9217	0.9222	0.9228	0.9233	0.9238	1	2	2	3	3	4	4	5	5	6
8.4	0.9243	0.9248	0.9253	0.9258	0.9262	0.9267	0.9272	0.9278	0.9283	0.9288	1	2	2	3	3	4	4	5	5	6
8.5	0.9294	0.9299	0.9304	0.9309	0.9314	0.9318	0.9323	0.9328	0.9333	0.9338	1	2	2	3	3	3	4	4	5	5
8.6	0.9345	0.9349	0.9354	0.9359	0.9364	0.9369	0.9374	0.9378	0.9383	0.9388	1	2	2	3	3	3	4	4	5	5
8.7	0.9395	0.9400	0.9405	0.9409	0.9414	0.9419	0.9424	0.9428	0.9433	0.9438	1	2	2	3	3	3	4	4	5	5
8.8	0.9445	0.9449	0.9454	0.9459	0.9464	0.9468	0.9473	0.9478	0.9482	0.9487	1	2	2	2	3	3	4	4	4	5
8.9	0.9494	0.9499	0.9503	0.9508	0.9513	0.9517	0.9522	0.9527	0.9531	0.9536	1	2	2	2	3	3	4	4	4	5
9.0	0.9542	0.9546	0.9551	0.9556	0.9560	0.9565	0.9570	0.9574	0.9579	0.9583	1	2	2	2	3	3	4	4	4	5
9.1	0.9586	0.9590	0.9595	0.9600	0.9604	0.9609	0.9613	0.9618	0.9622	0.9627	1	1	2	2	3	3	3	4	4	5
9.2	0.9630	0.9634	0.9639	0.9643	0.9648	0.9652	0.9656	0.9661	0.9665	0.9700	1	1	2	2	2	3	3	4	4	4
9.3	0.9685	0.9689	0.9694	0.9698	0.9703	0.9707	0.9712	0.9716	0.9721	0.9725	1	1	2	2	2	3	3	3	4	4
9.4	0.9730	0.9734	0.9739	0.9743	0.9747	0.9752	0.9756	0.9761	0.9765	0.9769	1	1	2	2	2	3	3	3	4	4
9.5	0.9772	0.9777	0.9781	0.9786	0.9790	0.9794	0.9798	0.9803	0.9807	0.9811	1	1	2	2	2	3	3	3	4	4
9.6	0.9814	0.9819	0.9823	0.9827	0.9831	0.9835	0.9840	0.9844	0.9848	0.9852	1	1	2	2	2	3	3	3	4	4
9.7	0.9856	0.9860	0.9864	0.9868	0.9872	0.9876	0.9880	0.9884	0.9888	0.9892	1	1	2	2	2	3	3	3	4	4
9.8	0.9895	0.9900	0.9904	0.9908	0.9912	0.9916	0.9920	0.9924	0.9928	0.9932	1	1	2	2	2	3	3	3	4	4
9.9	0.9937	0.9941	0.9945	0.9949	0.9953	0.9957	0.9961	0.9965	0.9969	0.9973	1	1	2	2	2	3	3	3	4	4

Example: log₁₀ 9.592 = 0.9819

HOW TO MAKE & USE COMMON LOG TABLES (Form 2 friendly)

Common Logarithm (base-10) Tables — How to BUILD and USE

Designed for Form 2 • Very simple language • English + Kiswahili

1) What is a “common log”?

English: It is a number that tells how many tens make another number. We write log x or log₁₀x.

Kiswahili: Ni nambari inayoonyesha ni mara ngapi tunazidisha kwa 10 ili kupata nambari nyingine. Tunaandika log x au log₁₀x.

2) Characteristic & Mantissa

Characteristic = whole-number part (can be negative). Mantissa = decimal part (always between 0 and 1). Example: log 250 = 2.3979 → characteristic = 2, mantissa = 0.3979.

Characteristic ni sehemu ya mzima (inaweza kuwa hasi). Mantissa ni sehemu ya desimali (kati ya 0 na 1). Mfano: log 250 = 2.3979 → characteristic = 2, mantissa = 0.3979.

Your printed table stores ONLY the mantissa for 1.00–9.99.

3) Table layout

Rows: 1.0, 1.1, 1.2, …, 9.9
Columns: 0–9 → gives 1.20, 1.21, … etc.
Mean Differences (Add): tiny numbers that adjust for a fourth digit.

Safu (rows), nguzo (columns), na “Mean Differences” = marekebisho madogo kwa tarakimu ya nne.

A. Super-easy rules you will memorize (no advanced math)

A1) Laws to remember

Add for ×: log(AB)=logA + logB
Ukizidisha, ongeza logi: log(AB)=logA + logB
Subtract for ÷: log(A/B)=logA − logB
Ukiigawa, toa logi: log(A/B)=logA − logB
Powers: log(Aⁿ)=n·logA • Roots: log(ⁿ√A)=logA / n
Nguli/Mizizi: log(Aⁿ)=n·logA, log(ⁿ√A)=logA / n

A2) Seed values to memorize (4 d.p.)

log2 = 0.3010

log3 = 0.4771

log4 = 0.6021

log5 = 0.6990

log6 = 0.7782

log7 = 0.8451

log8 = 0.9031

log9 = 0.9542

Hizi nane ukizikamata kichwani, unaweza kujenga karibu jedwali lote.

A3) “+0.01 step rule” (very handy for rows)

When x goes up by 0.01, the mantissa grows by about Δ ≈ 0.004343 / x. (Just divide 0.004343 by the current row number.)

Ukitoka x kwenda x+0.01, ongezeko la mantissa takribani Δ ≈ 0.004343 / x. Gawa tu 0.004343 kwa thamani ya safu x.

Example: around row x=1.50, add about 0.004343/1.50 ≈ 0.0029 each column step.
Mfano: karibu na x=1.50, ongeza ~0.0029 kwa kila kolamu (1.50→1.51→1.52…).

A4) Mean Differences (Add)

They tell how much to add for a fourth digit. If you are at 2.73 and the next digit is 6 (so 2.736), add the small number listed under 6 in the Mean Difference block of that row.

Zinaonyesha kiasi cha kuongeza kwa tarakimu ya nne. Ukiwa 2.73 na tarakimu inayofuata ni 6 (yaani 2.736), ongeza namba ndogo iliyo chini ya 6 kwenye sehemu ya “Mean Differences”.

B. HOW TO USE a common-log table (step by step)

Normalize the number so it sits between 1 and 10: write N = s × 10ᵏ with 1 ≤ s < 10.

Badilisha namba iwe N = s × 10ᵏ ambapo 1 ≤ s < 10.

Characteristic is k. (If the number is less than 1, k is negative.)

Characteristic ni k. (Chini ya 1 → hasi.)

Mantissa: row = first two significant digits (with a decimal); column = third digit; for a 4th digit add the Mean Difference.

Mantissa: safu = tarakimu mbili za mwanzo, kolamu = tarakimu ya tatu; ya nne tumia Mean Difference.

Add them: log N = k + mantissa.

Ongeza: log N = k + mantissa.

Quick example: Find log 9.592.
Row 9.5, column 9 gives 0.9818 (approx). 4th digit is “2” → add MD under 2 → 0.9819. Characteristic 0 → answer 0.9819.
Mfano wa haraka: Safu 9.5, kolamu 9; ongeza MD ya “2” → 0.9819. Hivyo jibu 0.9819.

C. HOW TO BUILD the table yourself (Form-2 friendly plan)

C1) Build anchor points (easy to memorize)

Memorize the 8 seed logs in section A2. From them you also get:
- log 1.5 = log3 − log2 = 0.4771 − 0.3010 = 0.1761
- log 2.5 = log5 − log2 = 0.6990 − 0.3010 = 0.3980 (≈ 0.3979 in tables)
- log 7.5 = log(15/2) = log15 − log2 = (0.1761+0.9999) − 0.3010 ≈ 0.8751 (close to 0.8751)
Kumbuka hizo 8; kisha tumia kuzidisha/kugawa kupata 1.5, 2.5, 3.5, … kama mifano hapo juu.

C2) Fill a whole row with the “+0.01 step rule”

Example: Row 1.5 (build 1.50 → 1.59)

Start value: log 1.50 = 0.1761.
Step size: Δ ≈ 0.004343 / 1.50 ≈ 0.0029.
Add repeatedly (update x each time). Round to 4 dp:
- 1.51: 0.1761 + 0.0029 ≈ 0.1790
- 1.52: +0.0029 ≈ 0.1819
- … continue to get 1.59 ≈ 0.2014

These match the printed table very closely.

Example: Row 8.0 (8.00 → 8.09)

Anchor: log 8 = 0.9031.
Step: Δ ≈ 0.004343 / 8.00 ≈ 0.00054.
Add up to column 9: 0.9031, 0.9036, 0.9042, … → matches standard row 8.0.

Kila mara zungusha hadi sehemu 4 za desimali.

C3) Make the “Mean Differences (Add)”

For any row entry (say 1.53 to 1.54), find the change in mantissa between adjacent columns (e.g., 0.1818 → 0.1847 = increase of 0.0029 = 29 in the 4th decimal place). Distribute that increase across digits 1–9. A quick way is:

Step in last place ≈ round(10⁴ × [log(x+0.01) − log(x)])
MD for digit d = round(d × step) − round((d−1) × step)

Tafuta tofauti kati ya kolamu mbili zilizo karibu; geuza kuwa “tarakimu ya mwisho” (mfano 0.0029 → 29). Gawa sawasawa kwa tarakimu 1–9 ukitumia kanuni hapo juu; haya ndiyo “Mean Differences”.

C4) Finish all rows

Repeat C1–C3 for rows 1.0 → 9.9. The step Δ slowly changes because x changes; recalc it or peek at the last two entries to keep your row smooth.
Rudia hatua hizo kwa safu zote. Kumbuka Δ hubadilika taratibu kadri x inavyobadilika.

D. Real-life, vivid uses (kujifunzia haraka)

1) Sound (Decibels)

Level L = 10 log(I/I₀). If sound intensity becomes 100×, add 10·log(100)=20 dB.

Ukiongeza nguvu mara 100, unaongeza 20 dB. Tumia jedwali kupata logi ndogo ndogo.

2) pH in Chemistry

pH = −log[H⁺]. If [H⁺]=3.2×10⁻⁴, log = −(−4 + mantissa of 3.2) = 4 − 0.5051 → pH ≈ 3.49.

Mantissa ya 3.2 (safu 3.2) kisha ongeza characteristic 4 → pH ≈ 3.49.

3) Fast multiplication

To multiply 2.73 × 6.8: add logs (0.4362 + 0.8325 ≈ 1.2687) → antilog gives ≈ 18.6.

Zidisha kwa kujumlisha logi kisha rudi kwenye namba (antilog) → ≈ 18.6.

4) Growth & interest

Estimate 1.07¹⁰: 10·log(1.07) ≈ 0.2938 → antilog ≈ 1.96.

Ukuaji wa 7% kwa miaka 10 ≈ 1.96 (karibu mara mbili).

E. Practice drills (for memory & speed)

Recall seeds daily: 0.3010, 0.4771, 0.6021, 0.6990, 0.7782, 0.8451, 0.9031, 0.9542. Zikariri kwa sauti.
Row builder: pick a row (e.g., 2.3). Use an anchor near it (2=0.3010, 2.5≈0.3980) then the +0.01 step rule to fill columns 0–9.
Mean differences: compute one cell’s step (difference × 10⁴), split across digits 1–9.
Use the table: do 5 quick logs and 5 quick antilogs daily.

F. Pocket cheatsheet

Normalize: N = s × 10ᵏ, 1 ≤ s < 10
Row = first 2 significant digits, Column = 3rd, add MD for 4th
log N = k + mantissa
Antilog: match mantissa in table → place decimal using k
Build rows: start from anchor, add Δ ≈ 0.004343/x, round to 4 d.p.

Jenga safu: anza na thamani ya karibu, ongeza Δ ≈ 0.004343/x, zungusha hadi sehemu 4.

Reference Book: N/A

Author name: SIR H.A.Mwala Work email: biasharaboraofficials@gmail.com
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Mathematical TablesCommon Logarithms of Numbers (Page 1)

Mathematical TablesCommon Logarithms of Numbers (Page 2)

Mathematical TablesCommon Logarithms of Numbers (Page 3)

Common Logarithm (base-10) Tables — How to BUILD and USE

1) What is a “common log”?

2) Characteristic & Mantissa

3) Table layout

A. Super-easy rules you will memorize (no advanced math)

A1) Laws to remember

A2) Seed values to memorize (4 d.p.)

A3) “+0.01 step rule” (very handy for rows)

A4) Mean Differences (Add)

B. HOW TO USE a common-log table (step by step)

C. HOW TO BUILD the table yourself (Form-2 friendly plan)

C1) Build anchor points (easy to memorize)

C2) Fill a whole row with the “+0.01 step rule”

C3) Make the “Mean Differences (Add)”

C4) Finish all rows

D. Real-life, vivid uses (kujifunzia haraka)

1) Sound (Decibels)

2) pH in Chemistry

3) Fast multiplication

4) Growth & interest

E. Practice drills (for memory & speed)

F. Pocket cheatsheet

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Mathematical Tables
Common Logarithms of Numbers (Page 1)

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Common Logarithms of Numbers (Page 2)

Mathematical Tables
Common Logarithms of Numbers (Page 3)