Determination of Logarithm and Antilogarithm

Finding a logarithm from a log table. A four-figure logarithm table lists the mantissae of digit strings. To find $\log N$:

1. Write down the characteristic using the rule of §4 (this comes from the size of the number, not the table).
2. Ignore the decimal point and read the mantissa of the significant digits from the table: the first two significant digits select the row, the third digit selects the main column, and the fourth digit is added through the mean-difference column by interpolation (§6).
3. Combine: $\log N = \text{characteristic} \,.\, \text{mantissa}$.

For example (worked fully in Worked Example 8), the digit string 4567 has mantissa $0.6596$, so $\log 45.67 = 1.6596$, $\log 4567 = 3.6596$ and $\log 0.4567 = \bar{1}.6596$ — the same mantissa throughout, only the characteristic changing.

Antilogarithm — the reverse operation. The antilogarithm of a number $x$ is the number whose logarithm is $x$; that is, $\text{antilog}(x) = 10^{x}$. It undoes a logarithm, so if $\log N = x$ then $N = \text{antilog}(x)$. To find an antilog from an antilog table:

1. Read the antilog table using only the mantissa (the decimal part) of $x$ — the row/column/mean-difference procedure is identical to the log table, and returns a four-digit number.
2. Use the characteristic of $x$ to fix the decimal point: a characteristic of $c$ (for $c \geq 0$) places the decimal point so that the result has $(c+1)$ digits before the point; a negative (barred) characteristic $\bar{c}$ produces a number less than 1 with $(c-1)$ zeros immediately after the decimal point.

For example (Worked Examples 9 and 10), $\text{antilog}(2.6596) = 456.7$ (the mantissa $0.6596$ gives the digits 4567; characteristic 2 gives 3 digits before the point), whereas $\text{antilog}(\bar{2}.6596) = 0.04567$ (same digits, but characteristic $\bar{2}$ places the value below 1 with one leading zero). …