How Do You Calculate Error Propagation for Logarithmic Functions?

[prize fight]

1. Homework Statement
Estimate the absolute and relative standard deviations of the following calculations. The number in parentheses is the standard deviation of the preceding value.

a) z=5.64(s=0.14)*log(138)(s=3)

2. Homework Equations
Sx/x =SQRT((Sp/P)2+(Sq/q)2+(Sr/R)2

Sx=0.434(Sp/P)

3. The Attempt at a Solution

I thought of two ways to go about the problem I am not sure which way is correct here are both attempts:

log(138)= 2.139
Z=12.069
SZ=SQRT((0.14/5.64)2+(3.0/2.139)2)=1.40
Z=12.07 LaTeX Code: \\pm 1.40

RSD=(1.4/12.07)*100 =11.6%

Or my other attempt:

Z=12.07
(0.14/5.64)2+(0.434*(3.0/138))=0.01
Z=12.07LaTeX Code: \\pm 0.01
RSD=(0.01/12.07)*100=0.08%

Are either of these ways correct? Any help would be appreciated. Thanks.

 

[LowlyPion]

1. Homework Statement
Estimate the absolute and relative standard deviations of the following calculations. The number in parentheses is the standard deviation of the preceding value.

a) z=5.64(s=0.14)*log(138)(s=3)

2. Homework Equations
Sx/x =SQRT((Sp/P)2+(Sq/q)2+(Sr/R)2

Sx=0.434(Sp/P)

3. The Attempt at a Solution

I thought of two ways to go about the problem I am not sure which way is correct here are both attempts:

log(138)= 2.139
Z=12.069
SZ=SQRT((0.14/5.64)2+(3.0/2.139)2)=1.40
Z=12.07 LaTeX Code: \\pm 1.40

RSD=(1.4/12.07)*100 =11.6%

Or my other attempt:

Z=12.07
(0.14/5.64)2+(0.434*(3.0/138))=0.01
Z=12.07LaTeX Code: \\pm 0.01
RSD=(0.01/12.07)*100=0.08%

Are either of these ways correct? Any help would be appreciated. Thanks.

I would choose a different approach. For:

z=5.64(s=0.14)*log(138)(s=3)

Certainly the RSS of the relative uncertainties is a good method. But in that regard I would prefer to treat the relative error of the 3/138 as really the relative uncertainty of the range of Log(138 ±3) which looks to me more like 2.14±.01, because that is the effect on the final result, as opposed to the 3/138.

Then I would choose to take the RSS of these relative terms according to the product rule.

((.14/5.64)² + (.01/2.14)²)^1/2

And calculate the absolute uncertainty from that expression.