Using four-figure tables · Logarithms

Most questions now say “without using tables”, and a calculator does the rest. A few still ask you to “use mathematical tables”, and then the working must show the table steps. This lesson is for those questions.

The logarithm of a number

Every log has two parts:

the characteristic, the whole-number part: the power of 10 when the number is in standard form;
the mantissa, the decimal part: read from the table, and always positive.

For example, take $75.9$ :

in standard form it is $7.59 \times 10^1$ , so the characteristic is 1;
the table entry for 7.59 is $.8802$ , which is the mantissa;
so $\log 75.9 = 1.8802$ .

Four-figure tables: logarithmsPick a number

N	0	1	2	3	4
39	5911	5922	5933	5944	5955
40	6021	6031	6042	6053	6064
41	6128	6138	6149	6160	6170

4.000 × 10³standard form3characteristic (the power of 10).6021mantissa: 60213.6021log 4000

In standard form, 4000 = 4.000 × 10³, so the characteristic is 3. Look up 40 in the first column, go across to column 0. The mantissa is always positive; only the characteristic can carry a bar.

To read the table:

the row is the first two figures (75);
the column is the third figure (9);
if there is a fourth figure, add the mean difference for it, from the small columns on the right.

A printed table has ten main columns, 0 to 9, and nine mean-difference columns headed 1 to 9. The board shows only the part you need.

Numbers less than 1: bar notation

$0.04735 = 4.735 \times 10^{-2}$ , so the characteristic is $-2$ . The mantissa stays positive, so the minus sign is written above the characteristic, as a bar:

\begin{aligned} \log 0.04735 &= \bar{2}.6754 \\ &= -2 + 0.6754 \end{aligned}

Bar notation2̄.6754: start at −2, then go forward 0.6754

Antilogarithms: back to the number

Look up the mantissa in the antilog table (row, column, mean difference) to get the figures. Then use the characteristic to place the decimal point. For $\bar{2}.6749$ :

the antilog table gives the figures 4731 for $.6749$ ;
the characteristic $\bar{2}$ means $\times 10^{-2}$ ;
so the number is $4.731 \times 10^{-2} = 0.04731$ .

Calculating with logs

To	do this to the logs
multiply	add them
divide	take them away
find a power	multiply the log by the power
find a root	divide the log by the root

Then take the antilog of the result. Set the working out in a table.

Worked example · NECO 2024

NECO 2024 · Paper 2 · Q6 (a)

Use mathematical tables to evaluate $\dfrac{\sqrt[4]{4000} \times 75.9}{5.61 \times 4.39}$ .

The fourth root
$\log 4000 = 3.6021$ . Divide by 4 for the fourth root: $0.9005$ .
Think first. How do you take a fourth root with logs?
The numerator
$\log 75.9 = 1.8802$ . Add: $0.9005 + 1.8802 = 2.7807$ .
The denominator
$\log 5.61 = 0.7490$ and $\log 4.39 = 0.6425$ . Add: $1.3915$ .
Divide and take the antilog
- Top minus bottom: ${2.7807 - 1.3915 = 1.3892}$ .
- The antilog of ${.3892}$ gives the figures 2450.
- The characteristic 1 means ${\times 10^1}$ , so the answer is about 24.50.

Dividing a bar characteristic

To halve $\bar{1}.5540$ , you can’t just halve the $\bar{1}$ : half of $-1$ isn’t a whole number. So first borrow 1 to make the characteristic even:

borrow 1, since ${-1 = -2 + 1}$ : ${\bar{1}.5540 = \bar{2} + 1.5540}$ ;
halve each part: ${\bar{1} + 0.7770}$ ;
so half of ${\bar{1}.5540}$ is ${\bar{1}.7770}$ .

Your turn

WAEC 2025 · Paper 2 · Q6 (a)✱✱

(a)
Using mathematical tables, find: (i) $2\sin63.35^\circ$ ; (ii) $\log(\cos44.74^\circ)$ ; (iii) $k$ , given that $\log k - \log(k - 2) = \log 5$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)(i)

From the tables, $\sin 63.35^\circ \approx 0.8938$ , so $2\sin 63.35^\circ \approx 1.788$ .

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