Definition:Log Tables

Definition

Log tables are a set of reference tables for calculation of common logarithms of real numbers.

They consist of a set of lookup tables for the mantissa of the common logarithm of the number in question.

It is then the task of the user to provide the characteristic by inspection.

Examples

Four-Figure 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}$

Also see

• Results about log tables can be found here.

Historical Note

Before the advent of inexpensive electronic calculators, the use of log tables was a standard technique for multiplication of multi-digit numbers.

The technique consisted of:

$(1): \quad$ Use the log tables to find the logarithms of $x$ and $y$ to obtain $\log_{10} x$ and $\log_{10} y$.

$(2): \quad$ Add those logarithms to obtain $\log_{10} x y$.

$(3): \quad$ Use the log tables to find the antilogarithm of $\log_{10} x y$.

The result is $x y$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): logarithm (log)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): logarithm (log)