Q.Find $\log 45.67$ using a four-figure logarithm table, showing the mean-difference (interpolation) step for the fourth significant digit.

The characteristic comes from the size of the number and the mantissa is read from the table for the digits 4567, with the fourth digit added by interpolation.

Step 1 — Characteristic. The number 45.67 has 2 digits before the decimal point, so its characteristic is $2 - 1 = 1$.

Step 2 — Mantissa from the table. Ignore the decimal point and work with the digit string 4567. From the log table, the row 45, column 6 (first three significant digits, 456) gives the base mantissa

$\log 4.560 = 0.6590.$ …

Cross-check by the quotient law using neighbouring whole numbers. $\log 45 = 1.6532$ and $\log 46 = 1.6628$; interpolating linearly for 45.67 gives $1.6532 + 0.67\times(1.6628 - 1.6532) = 1.6532 + 0.67\times0.0096 = 1.6532 + 0.0064 …

The most frequent error is adding the mean difference to the characteristic instead of the mantissa, or reading the characteristic from the table (the table only supplies the mantissa — the characteristic must be worked out separately from …