# Thread: Deriving logarithmic error equation

1. ## Deriving logarithmic error equation

Hi guys, wonder if any of you can help me clarify something. Got a deadline coming up and have been asked to explain an equation I've used in my work ... I've got a derivation from the source I got the equation from, but I don't understand one of the steps ... here goes ....

$m\pm \sigma_m = constant - 2.5\log(S\pm N)$
$m\pm \sigma_m = constant - 2.5\log\left[S\left(1\pm \frac{N}{S}\right)\right]$
$m\pm \sigma_m = constant - 2.5\log S - 2.5\log\left(1+ \frac{N}{S}\right)$
Therefore
$\sigma_m = \pm 2.5\log \left(1 + \frac{1}{S/N}\right)$

I'm just not sure what's happening with the $\pm$ disappearing, changing places etc, so if anyone could show me how you get from line 2 to 3 and 3 to 4 I'd be very grateful!

Cheers,
Rai

2. Originally Posted by Rai
Hi guys, wonder if any of you can help me clarify something. Got a deadline coming up and have been asked to explain an equation I've used in my work ... I've got a derivation from the source I got the equation from, but I don't understand one of the steps ... here goes ....

$m\pm \sigma_m = constant - 2.5\log(S\pm N)$
$m\pm \sigma_m = constant - 2.5\log\left[S\left(1\pm \frac{N}{S}\right)\right]$
$m\pm \sigma_m = constant - 2.5\log S - 2.5\log\left(1+ \frac{N}{S}\right)$
Therefore
$\sigma_m = \pm 2.5\log \left(1 + \frac{1}{S/N}\right)$

I'm just not sure what's happening with the $\pm$ disappearing, changing places etc, so if anyone could show me how you get from line 2 to 3 and 3 to 4 I'd be very grateful!

Cheers,
Rai
From line 2 to line 3,
The reason the negative N/S is not included anymore could be that N is geater than S.
If N > S, then N/S is greater than 1.0. Then (1 -N/S) is negative.
There are no logarithms for negative quantities.

From line 3 to line 4,
The reappearance of the +,- outside of the logarithm is due to the +,- of the sigma_m in line 3.
In line 4, the sigma_m is only positive.

3. Thanks a lot for that. Of course, the reason for the $\pm$ reappearing in the 4th line is obvious now you've pointed it out! Hmmm N isn't likely to be larger than S in many cases (the symbols stand for noise and signal and in the context the equation is used in, the signal measured is usually much larger than the noise measurement). I've got a horrible feeling it's just been 'fudged' ... I shall have to have another think ...
Thanks again!