Q.Find the antilogarithm of $\bar{2}.6596$ (that is, characteristic $-2$, mantissa $0.6596$).

The mantissa gives the same significant digits as before, but the negative (barred) characteristic now places the number below 1 with a leading zero.

The mantissa 0.6596 again yields the digits 4567, and a characteristic of $\bar{2}$ places the value below 1 with one leading zero, giving 0.04567.

Step 1 — Digits from the mantissa. The mantissa is again $0.6596$, so the antilog table returns the same four significant digits, 4567 (exactly as in Worked Example 9 — the mantissa alone determines the digits, never the characteristic).

Step 2 — Place the decimal point using the negative characteristic. The characteristic is $\bar{2} = -2$. A number with a negative characteristic is less than 1; a characteristic of $\bar{c}$ produces $(c - 1)$ zeros immediately after the decimal point before the first significant digit. Here $c = 2$, so there is $2 - 1 = 1$ leading zero:

Cross-check by taking the logarithm back. The number 0.04567 is less than 1 with one leading zero after the point, so its characteristic is $-[1+1] = \bar{2}$ (§4), and its digit string 4567 has mantissa 0.6596; hence $\log 0.04567 = \bar{2}.6596$. Taking the log of the answer returns the original input, confirming the result. Note it is exactly the Worked Example 9 answer (456.7) divided by $10^{4}$, matching the change in characteristic from 2 to $\bar{2}$.

The classic mistake here is arithmetic on the negative logarithm: converting $\bar{2}.6596$ to $-2.6596$ (which is wrong — it actually equals $-2 + 0.6596 = -1.3404$) and then reading the antilog of a wrong value. Keep the characteristic and mantissa separate; the mantissa going into the table is always the positive $0.6596$.