Log Tables/Examples/Four-Figure Tables

Example of Log Tables

The following is an extract of a set of log tables that express the mantissa of the common logarithm of a given number to $4$ significant figures:

$\begin {array} {|r|c|c|c|c|c|c|c|c|c|c|rrr|rrr|rrr|} \hline & \mathbf 0 & \mathbf 1 & \mathbf 2 & \mathbf 3 & \mathbf 4 & \mathbf 5 & \mathbf 6 & \mathbf 7 & \mathbf 8 & \mathbf 9 & \mathbf 1 & \mathbf 2 & \mathbf 3 & \mathbf 4 & \mathbf 5 & \mathbf 6 & \mathbf 7 & \mathbf 8 & \mathbf 9 \\ \hline \mathbf {10} & \cdotp 0000 & \cdotp 0043 & \cdotp 0086 & \cdotp 0128 & \cdotp 0170 & \cdotp 0212 & \cdotp 0253 & \cdotp 0294 & \cdotp 0334 & \cdotp 0374 & 4 & 8 & 12 & 17 & 21 & 25 & 29 & 33 & 37 \\ 11 & \cdotp 0414 & \cdotp 0453 & \cdotp 0492 & \cdotp 0531 & \cdotp 0569 & \cdotp 0607 & \cdotp 0645 & \cdotp 0682 & \cdotp 0719 & \cdotp 0755 & 4 & 8 & 11 & 15 & 19 & 23 & 26 & 30 & 34 \\ 12 & \cdotp 0792 & \cdotp 0828 & \cdotp 0864 & \cdotp 0899 & \cdotp 0934 & \cdotp 0969 & \cdotp 1004 & \cdotp 1038 & \cdotp 1072 & \cdotp 1106 & 3 & 7 & 10 & 14 & 17 & 21 & 24 & 28 & 31 \\ \vdots \\ 44 & \cdotp 6435 & \cdotp 6444 & \cdotp 6454 & \cdotp 6464 & \cdotp 6474 & \cdotp 6484 & \cdotp 6493 & \cdotp 6503 & \cdotp 6513 & \cdotp 6522 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 45 & \cdotp 6532 & \cdotp 6542 & \cdotp 6551 & \cdotp 6561 & \cdotp 6571 & \cdotp 6580 & \cdotp 6590 & \cdotp 6599 & \cdotp 6609 & \cdotp 6618 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 46 & \cdotp 6628 & \cdotp 6637 & \cdotp 6646 & \cdotp 6656 & \cdotp 6665 & \cdotp 6675 & \cdotp 6684 & \cdotp 6693 & \cdotp 6702 & \cdotp 6712 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 7 & 8 \\ \vdots \\ 97 & \cdotp 9868 & \cdotp 9872 & \cdotp 9877 & \cdotp 9881 & \cdotp 9886 & \cdotp 9890 & \cdotp 9894 & \cdotp 9899 & \cdotp 9903 & \cdotp 9908 & 0 & 1 & 1 & 2 & 2 & 3 & 3 & 4 & 4 \\ 98 & \cdotp 9912 & \cdotp 9917 & \cdotp 9921 & \cdotp 9926 & \cdotp 9930 & \cdotp 9934 & \cdotp 9939 & \cdotp 9943 & \cdotp 9948 & \cdotp 9952 & 0 & 1 & 1 & 2 & 2 & 3 & 3 & 4 & 4 \\ 99 & \cdotp 9956 & \cdotp 9961 & \cdotp 9965 & \cdotp 9969 & \cdotp 9974 & \cdotp 9978 & \cdotp 9983 & \cdotp 9987 & \cdotp 9991 & \cdotp 9996 & 0 & 1 & 1 & 2 & 2 & 3 & 3 & 3 & 4 \\ \hline & \mathbf 0 & \mathbf 1 & \mathbf 2 & \mathbf 3 & \mathbf 4 & \mathbf 5 & \mathbf 6 & \mathbf 7 & \mathbf 8 & \mathbf 9 & \mathbf 1 & \mathbf 2 & \mathbf 3 & \mathbf 4 & \mathbf 5 & \mathbf 6 & \mathbf 7 & \mathbf 8 & \mathbf 9 \\ \hline \end {array}$

They are used as follows.

Let $x$ be the number whose mantissa $m$ of the common logarithm is sought.

Locate the row of the log table whose leftmost entry is the two most significant digits of $x$.

Locate the column of the left hand set of columns whose heading is the $3$rd most significant digit of $x$.

This provides a $4$ digit number $m'$.

Locate the column of the right hand set of columns whose heading is the $4$th most significant digit of $x$.

This provides an adjustment which is to be added to the least significant digits of $m'$ to obtain $m$.

For example, let us seek the mantissa of the common logarithm of $4 \cdotp 526$.

We locate the row whose leftmost entry is $45$.

We locate the column of the left hand set headed by $2$.

This we see is $\cdotp 6551$.

We locate the column of the right hand set headed by $6$.

This contains $6$.

We add $6$ to $6551$ to obtain $6557$.

Hence the mantissa of the common logarithm of $4 \cdotp 526$ is $0 \cdotp 6557$.

Sources

• 1947: C. Godfrey and A.W. Siddons: Four-Figure Tables: Logarithms