How do you properly add logarithms with negative characteristics by hand?

[gerid21]

Homework Statement

Calculate $890\times12.34\times0.0637$ using logarithms of base 10

Relevant Equations

Log tables, calculator

From the log tables:
$log(890) = 2.9494, \space log(12.34)=1.0913, \space log(0.0637)=\bar{2}.8041$

I calculate by hand:
$\begin{array}{r} &2.9494\\ +&1.0913\\&\bar{2}.8041\\\hline &2.8448 \end{array}$

Thus:
$log^{-1}(2.8448) \approx 699.6 \space$

Which is the correct answer.

Now I know $log(0.0637)=\bar{2}.8041=-2+0.8041=-1.1959$

Yet if I did

$\begin{array}{r} &2.9494\\ +&1.0913\\&-1.1959\\\hline &3.2366 \end{array}$

$log^{-1}(3.2366)\approx1724\space$ This is obviously wrong.

But if I were to do the problem solely with a calculator, I have to plug in $log(0.0637)= -1.1959\space$ and not $-2.8041$ to get the correct answer. I'm confused at what's going on. I know it has something to do with the negative characteristic of the logarithm. But then if I'm adding it by hand, why doesn't $-1.1959$ work?

 

[DrClaude]

$\begin{array}{r} &2.9494\\ +&1.0913\\&-1.1959\\\hline &3.2366 \end{array}$

When I add these numbers, I get 2.8448.

 

[gerid21]

When I add these numbers, I get 2.8448.

So essentially $$\bar{2}.8041$$ means only the number under the bar is negative and the rest is positive whereas $$-1.1959$$ means the entire number before and after the decimal is negative?

 

[DrClaude]

So essentially $$\bar{2}.8041$$ means only the number under the bar is negative and the rest is positive whereas $$-1.1959$$ means the entire number before and after the decimal is negative?

The latter is the standard mathematical notation. I'm not that familiar with log tables, but it seems that the former is the common notation used in those tables:
https://www.wikiwand.com/en/Common_logarithm

 

[epenguin]

Log tables, how quaint!

So essentially $$\bar{2}.8041$$ means only the number under the bar is negative and the rest is positive whereas $$-1.1959$$ means the entire number before and after the decimal is negative?

Yes, and probably you knew and then forgot, because log tables only go from 1 to 9.999 which far from includes all numbers, so you would have to have known that to look up and use. Maybe this not so drilled into students these days, and that's no loss.

(The two parts are called 'mantissa' and 'exponent' by the way, to jog your memory or if you want to look them up.)

 

[pbuk]

I'm not that familiar with log tables, but it seems that the former is the common notation used in those tables:

Yes - otherwise they would be twice as long!

 

[epenguin]

Yes - otherwise they would be twice as long!

They could be millions of times as long and it wouldn't be enough!

 

[Merlin3189]

Now I know $log(0.0637)=\bar{2}.8041=-2+0.8041=-1.1959$

Yet if I did

$\begin{array}{r} &2.9494\\ +&1.0913\\&-1.1959\\\hline &3.2366 \end{array}$

Strange you understand that $\bar{2}.8041=-2+0.8041$ and correctly add it up in that form, but don't realize that the more common $-1.1959 = -1 -0.1959$ and add that up wrongly.

But if you did it properly, it works
$\begin{array}{r} &2.9494\\ +&1.0913\\\hline &+4.0407\\&-1.1959\\\hline &+2.8448 \end{array}$

I wonder for which method you would do it right when you subtract to do a division?