## Logarithm Tables

### September 30, 2011

The requirement to always display mantissas accurate to within 1 in 10000 makes this exercise harder than it looks. The two tricks are rounding instead of truncating, and calculating the mean (average) differences instead of some lower-bound difference for the extra digit. Here’s the log table:

`(define (log-average-interp i k)`

(define (add-on j)

(- (* (log10 (/ (+ i (* j 10) k) 1000.0)) 10000)

(* (log10 (/ (+ i (* j 10)) 1000.0)) 10000)))

(inexact->exact (round (average (map add-on (range 10))))))

`(define (log-display-interp i)`

(display " ")

(do ((k 1 (+ k 1))) ((= 10 k))

(display " ")

(display (pr2 (log-average-interp i k)))))

`(define (log-display-line i)`

(do ((j 0 (+ j 10))) ((= 100 j))

(display " ")

(display (pr4 (log10 (/ (+ i j) 1000)))))

(log-display-interp i))

`(define (log-display-table)`

(display-header)

(do ((i 1000 (+ i 100))) ((= 10000 i))

(if (zero? (modulo i 1000)) (display-bar))

(display (/ i 100))

(display " ")

(log-display-line i)

(newline)))

The mean difference is calculated in the `log-average-interp`

function, where each fourth digit is calculated as the average amount of interpolation required over all ten possible third digits. Anti-logarithms are calculated similarly, but backwards:

`(define (alog-average-interp i k)`

(define (add-on j)

(- (* (expt 10 (+ 1 (/ (+ i (* j 10) k) 10000))) 100)

(* (expt 10 (+ 1 (/ (+ i (* j 10)) 10000))) 100)))

(inexact->exact (round (average (map add-on (range 10))))))

`(define (alog-display-interp i)`

(display " ")

(do ((k 1 (+ k 1))) ((= 10 k))

(display " ")

(display (pr2 (alog-average-interp i k)))))

`(define (alog-display-line i)`

(do ((j 0 (+ j 10))) ((= 100 j))

(display " ")

(display (pr4 (/ (expt 10 (+ 1 (/ (+ i j) 10000))) 100))))

(alog-display-interp i))

`(define (alog-display-table)`

(display-header)

(do ((i 1000 (+ i 100))) ((= 10000 i))

(if (zero? (modulo i 1000)) (display-bar))

(display (/ i 100))

(display " ")

(alog-display-line i)

(newline))))

We also need some auxiliary functions to handle the formatted printing, in addition to `range`

, `average`

and `log10`

from the Standard Prelude:

`(define (pr4 x)`

(let* ((x (inexact->exact (round (* x 10000))))

(s (number->string x)))

(if (< x 1000) (set! s (string-append "0" s)))

(if (< x 100) (set! s (string-append "0" s)))

(if (string x)))

(if (< x 10) (set! s (string-append " " s)))

s))

`(define (display-header)`

(display " ")

(do ((j 0 (+ j 1))) ((= j 10))

(display " ")

(display j))

(display " ")

(do ((j 1 (+ j 1))) ((= j 10))

(display " ")

(display j))

(newline))

`(define (display-bar)`

(display "-- ";)

(do ((j 0 (+ j 1))) ((= j 10))

(display " ") (display "----"))

(display " ")

(do ((j 1 (+ j 1))) ((= j 10))

(display " ") (display "--"))

(newline))

You can run the program at http://programmingpraxis.codepad.org/fiKOD0wV. The output is shown on the previous page.

I remember using tables 20 years ago and even 10 years ago. Using tables at the exam instead of or in addition to a calculator is still allowed in Denmark as far as I know, and it actually provides a benefit, because you can double check your results. I remember my previous mathematics teacher even taught us pupils to use a ruler and caliper, because it gave us a much better understanding of what logarithms were about.

I learned to use logarithms in school a long time ago (30 years or so). In my office, I have the 1957-58 edition of the CRC, which contains tables of logarithms to 4 digits, to 5 digits, logarithms of trig functions, etc. I haven’t used logarithm tables directly since school. However, I have several slide rules (which are based on logarithms) and use them on a regular basis.

Here is my python 3 code to produce the log and alog tables.

Mike: I still use my father’s slide rule. And I recently bought a 1.5″ diameter circular slide rule that hangs on my keychain — it’s already proven useful.

The bezel on my watch is a cicular slide rule, the pencil holder on my desk is a cylindrical one, and I have an E6B flight computer. I also have several of my fathers slide rules that date from the 50’s. My favorite is a credit card sized circular slide rule with a periodic table on the back and a pull out card with all sorts of mathmatical and physical constants, conversion formulas, etc.

When I served on a submarine we used ‘whiz wheels’ all the time. A skilled person with a whiz wheel could often do a computation faster then a person using a computer. We also practiced calculating targeting solutions in our heads.