What this number means: $\bar{2}.767$

Question

I faced this situation when I was solving this problem:

$\log(x) = 1.233 \Rightarrow colog (x) = ?$

The right answer was:

$\bar{2}.767$

Answer 1

With logarithms one sometimes uses a notation where the fractional part of a decimal number always counts positively.

With this notation $\overline 2.767$ means $(-2)+0.767$ or in other words $-1.233$.

This is useful especially when working with logarithm tables (which essentially nobody does nowadays when we all have pocket computers) because $10^{\overline 2.767} = 10^{-2}\cdot 10^{0.767}$ -- you can look up $10^{0.767}$ in a logarithm table and then just move the point to account for the integer part of the logarithm.

In contrast, to use $10^{-1.233}$ directly you would need to do $10^{-1}/10^{0.233}$, which would need a long division by the number you get from the logarithm table, and negate the practical point of using logarithms in the first place.

Answer 2

\begin{align} -1.233 &= -1 -0.233\\ &=-2+1-0.233\\ &=-2+0.767\\ &=\bar{2}.767 \end{align}

That is the meaning of the bar: $-2 + 0.767$.