Finding the absolute error for equation of total resistance

[AdrianMachin]

Homework Statement

Prove the equation for ΔR_C. (See the first attachment with the circuit schematic inside)

Homework Equations

(The equation for R_C, in both attachments)

The Attempt at a Solution

You can see my attempt at a solution in the second attachment. I'm stuck at the step shown in the picture. Maybe that needs a bit algebra, or my attempt was incorrect.

 

[BvU]

I already have difficulty with the first step: If $R_C = R_{//} + R_3$ then it certainly is not so that $\log R_C = \log R_{//} + \log R_3$ !
And then the $\Rightarrow$ is also unjustified.

I suggest you try to find $\Delta R_{//}$ on its own...

 

[mfb]

ln(a+b) and ln(a) + ln(b) are not the same (first -> second line and also within the second line) and the result is wrong.

 

[AdrianMachin]

I already have difficulty with the first step: If $R_C = R_{//} + R_3$ then it certainly is not so that $\log R_C = \log R_{//} + \log R_3$ !
And then the $\Rightarrow$ is also unjustified.

I suggest you try to find $\Delta R_{//}$ on its own...

Thanks a lot, dear BvU! I tried your suggestion and it worked!
Yes, I had a big mistake in my first attempt as you mentioned.

What do you mean by this:

And then the $\Rightarrow$ is also unjustified.

Is that because of my mistake in taking logarithms or it's generally not considered right to use in these equations?

 

[BvU]

Transition from first line in attachment to second line was in error.
The transition from before the $\Rightarrow$ to after was also in error, hence my remark.
$$\ln{R_1 R_2\over R_1+R_2} =\ln R_1+\ln R_2-\ln(R_1+R_2) $$