# SEM involving a logarithmic

1. Feb 16, 2016

### TheBoro76

Hi, this isn't exactly a homework question, but this seemed like the most appropriate place to put it.
1. The problem statement, all variables and given/known data
I have an equation in the form:
log(a)=log(b)+c.
I also have standard errors (SEMs) for b and c. I want to find the standard error for log(a) (i.e. log(a) +/- E(log(a)))

2. Relevant equations
I know the SEM of some quantity x, where x:=y+z, is given by Ex=sqrt(Ey^2+Ez^2)

3. The attempt at a solution
The problem is really trying to find the error of the log.

In high school I would have solved it by choseing the largest of:
abs(log(b+Eb)-log(b)) and abs(log(b-Eb)-log(b)), where Eb is the SEM of b.
if we let this be g then
E(log(a))=sqrt(g^2+Ec^2)

However, given I am doing uni research I am not sure whether this would be acceptable.

I have also considered making a Monte-Carlo simulation of the problem. Drawing random numbers from the distributions b~N(b,Eb) and c~N(c,Ec) and finding the mean and standard deviation of the simulation. However I would like to get an analytical solution.

Thanks in advance if anyone can help me out

2. Feb 16, 2016

### Staff: Mentor

When you have a function of several variables with errors a common approach is to take partial derivatives of the function with respect to each variable and then combine the errors in quadrature.

Say you have $f(a,b,c)$ with $a ± Δa$, $b ± Δb$, and $c ± Δc$. Then the total error is given by:

$$Δf = \sqrt{\left(\frac{\partial f}{\partial a}\right)^2 Δa^2 + \left(\frac{\partial f}{\partial b}\right)^2 Δb^2 + \left(\frac{\partial f}{\partial c}\right)^2 Δc^2 }$$

Do you remember how to differentiate the log function?