Understand Logarithm of Bar values manipulation step.

Currently I am learning Logarithm , but I can't understand the manipulation of the following Highlighted step how it comes

How the result come after after 1/2 (bar 1.7563)

Kindly guide me on the same.

Logarithm is broken up into two parts.

The first part is the integral part the the rest is the fractional part WHICH MUST BE POSITIVE.

The integral part takes care of the number of decimal places of the original number.

For example, In log(8432) = 3. 9259, 3 means the original number (ie. 8432) has 3 + 1 digits. Then, (bar 2). 7563 means -2 + 0.7563 = -1.2437

But -1.2437 means -1 – 0.2437 which is not an allowable format of a logarithm.

Therefore, we have to re-write as

– 1 – 0.2437 = – 1 – 0.2437 + 1 – 1

= –1 – 1 + 1 – 0.2437

= – 2 + (1 – 0.2437)

= - 2 + 0.7563

= (bar2. 7563)

Added info

Let $p = \frac {bar(1).7563}{2}$. After division, $p$ becomes $bar(0.5).37815$, which is NOT in the allowable format because the integral part is NOT integral any more.

The book makes the suggestion:- Convert $p$ to $\frac {bar(2) + 1.7563}{2}$ first.

Then, $p = \frac{ bar (2)}{2} + \frac {1.7563}{2} = bar (1) + 0.8782 = bar (1).8782$

There is also another (could-be more understandable) way to get around it:-

Step-1) First, obtain its numerical equivalent (but NOT in the allowable format).

∴ $p = \frac {-1 + 0.7563}{2} = \frac {– 0.2437}{2} = – 0.12185 = – 0.1218$ (approx.)

Step-2) Then, convert the above back to the standard form.

∴ $p = – 0.1218 = (– 1) + (1 + – 0.1218) = (– 1) + (0.8782) = bar (1).8782$.

Bar 1 is not divisible by 2 so you add another bar 1 to the characteristic part to make it divisible by 2 because you added bar 1, add 1 to the mantissa part and then divide the two parts separately and then add the results as shown below

Characteristic

bar 1 +bar 1 =bar 2 bar2 *1/2= bar1

Mantissa
0.7563+1=1.7563 1.7563*1/2=0.8782

Adding the characteristic and mantissa gives 1 bar+0.8782= 1 bar.8782