Just how important is precise resistor value?

[Phend]

You can figure log on any calculator with a square root key. Just take the square root of any number "eleven" times, subtract 1 and multiply by 889. Da.  On another note back in 77 at LSSC I had the pleasure of taking the LAST how to use a slide rule class. It was required. Why?  Calculators were "new tech" then, What if the batteries died !! (Solar power calculators were a decade away)

[PRR]

> the LAST how to use a slide rule class

My youngest brother not only took the last (high school) class, he took home the 7-foot slide rule we all had used in the classroom.

[Rob Strand]

My youngest brother not only took the last (high school) class, he took home the 7-foot slide rule we all had used in the classroom.

Does that mean the teacher always got bigger numbers than the students? 

[PRR]

> root of any number "eleven" times ... ... ... ... .... ... ... ... ...

Why eleven? Ah, this one answer Radio Shack did have:
https://www.sliderulemuseum.com/Calculators/Radio_Shack_EC-425_Instructions.pdf


This may be cribbed from Texas Instruments SR-10:
https://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/archv018.cgi?read=145192

[Rob Strand]

Why eleven?

I think the 11 is determined by the rounding errors based on the calculator's number of digits.

If you write the result more generally as,

           ln(x)  ~  (n-1) * ( x ^ (1/n) - 1)

it gets more accurate as n gets larger for exact calculations.
The other way to spin that is it is more accurate over a wider range of values.

The current approximation for 11 times square roots is n = 2048.
(If you want log10(x) divide by 2.302... etc.)

Interestingly, for reasonably large n the approximation has (near) zero error
for x ~ 7.39 over a wide range of n.
-------------------------------
EDIT:  It just occurred to me exactly why it's rounding.

If you take a number, say 10, and you keep taking square roots you end up with,

.
.
.
1.0011249      ; 5 information digits
1.0005623
1.0002811
1.0000141      ;3 information digits

With a fixed number of digits a point is reached where the important information in
the decimal part starts to fall off the end.    If you take too many square roots
and subtract off the 1 you will have little information left.

Not taking enough square roots (small n) has an approximation error.
Taking too many square roots (large n) has a (digit) truncation/rounding error.
There is a happy medium in between.

[Phend]

Hum....and x 889 ?

[Rob Strand]

Hum....and x 889 ?

There's two types of logs, one base 'e' and one base 10:
 
   ln(x)   and    log10(x)

log10(x)   = ln(x) / ln(10)  = ln(x) / 2.3026 ...

For (n-1) = 2047, we get the constant 2047 for ln(x)
and for that same n,  the constant for log10(x) is then,

2047 / 2.3026 ...  = 889

So, the 889 is for log10(x) and 2047 is for ln(x).

Both appear in PRR's radioshack image extract.

[EBK]

Hum....and x 889 ?

There's two types of logs, one base 'e' and one base 10:
 
   ln(x)   and    log10(x)

There are an infinite number of log bases.
E.g., log₂(256)=8

and

log_noodles(x)=log(x)/log(noodles) 

[Rob Strand]

lognoodles(x)=log(x)/log(noodles)

I know that but the general public only sees calculators and log tables with ln(x) and log10(x).   If you grew up with log tables you might think log10() is more common because it was used for hand calculations but for mathematicians ln(x) is far more common.    These days I only use log10() when doing dB calculations.

When you compute ln(x) on a computer it is very efficient to think in terms of log2(x) since 2^x is just a bit shift operation.

[EBK]

I know that ...

I'm not suggesting that you didn't.  I was just in a silly mood.  I suspect log(noodles) must be approximately this, by the way:

[Rob Strand]

I was just in a silly mood.  I suspect log(noodles) must be approximately this, by the way:

I'm very familiar with those logs

[Phend]

PRR, Rob, I am impressed. I got that "formula" years ago. Not being the math guy, I thought it was just a "novel".  I am told that a Vermont mathematicians favorite food is "Log Pi"

[Rob Strand]

PRR, Rob, I am impressed. I got that "formula" years ago. Not being the math guy, I thought it was just a "novel".  I am told that a Vermont mathematicians favorite food it "Log Pi"

We should thank you.   I haven't seen that log trick before.    (Approximating functions is something I've spent alot of time on.)