Re: Use of Inverse LogLog scales Part 5: Logarithms


Andreas Poschinger
 

Hi,

this part is now about the use for logarithms for numbers smaller than
1, that I've almost forgot.

I still teach Bode (though in my practical time never applied) with the
amplidude response be calculated by 20 log(U_a/U_e), with means that we
take the ratio of output to input, e.g. of voltages, take the decadic
logarithm of this ratio and multiply it in order to obtain so called
Decibel (dB).

Usually it is not necessary to actually calculate the logarithm, when
knowing that 6dB corresponds to the ratio 2:1, 3db=1/2*6dB to the ratio
sqrt(2:1), 2 dB to the ratio 1.26:1. By these all other values can be
derived, e.g. 11dB = 4*2dB+3dB and thus corresponding to (1.25)^4 *
sqrt(2) according to logarithmic calculation laws, which still can be
calculated on a Mannheim layout without any LL scales. Thus so far I
thought, that in my field logarithms were not really needed in slide
rule times.

In wikipedia I've however found that until the 70ies the Neper (Np)
calculated by A/Np=ln(U_a/U_e) was dominant above dB and thenin the
70ies it has changed. Does anybody know about that from own experience?

Neper would be simpler to be calculated on the sliderule, since we get
rid of 20 times and a base which is not the Euler number e.

Let's e.g. assume U_a=2.5V and U_e = 5V.

Using LL02:

Calculate 2.5/5 e.g. by C:5 to D:2.5; at C:10: result D:0.5
Cursor to 0.5 on LL02; result -0.693 on D (the - comes from e^-0.1x,
which es the expression calculated by LL02 from values X between 1 and
10 fed into D!)

Using LL2:

Calculate 5/2.5 e.g. by C:5 to D:2.5; at C:10: result D:2

Cursor to 2 on LL2; result -0.693 on D (the - comes from having inversed
the ratio!)

So we do not save time using LL2, we only can use a more direct recipe.

For the sake of completeness:

The LL scales can deliver logs of all kinds, so e.g. calculating
20*log(2) (decadic):

First cursor then C:1 (or for other cases C:10) to base 10 on LL3; at 2
on LL2: 0.301 on C;

C:1 to 3.01 on D; at C20: result 6.02 on D

That is why I e.g. do not understand that the Studio wastes space on
slide for an L scale, instead using this space e.g. for a second S scale.

Any other usage for (inverse) LL scales?

Best regards

Andreas

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