Error Propagation calculation

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
Homework Helper
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.