The Ln Scale

The Ln scale Robert Adams

1 Introduction Earlier in this International Meeting a paper by Richard Hughes has introduced a number of rules for electronic calculations. Many of the scales are alien to slide rule collectors and are purely for impedance and resonance calculations within electronic and electrical circuits. One scale that appears only on Pickett & Eckel, USA (Pickett) slide rules deserves closer inspection. It was one of the scales that went to the Moon as part of the Pickett 600 models Ref 3. The scale is useful and can be used in many areas of mathematics, physics and engineering. Why it was not common on slide rules remains an interesting question. The answer lies, I think, in a curious mix of patent protection, ingrained manufacturing systems and a lack of promotion. The scale I am referring to is the Ln scale, which is a linear scale, used along with the C and D scales for finding natural (base e) logarithms and exponentials. The following partial scan of a Pickett 515T shows the Ln scale at the bottom of the body.

Figure 1- Pickett N-515-T

Aluminium 308 x 38 x 2.5 mm open frame single line cursor (1965)

2 Log Scales There are a number of scales on common slide rules that are called “log” scales. For instance there is the L scale, a linear scale, used along with the C and D scales for finding base-10 logarithms and powers of 10. And then there are the LL scales commonly called log-log scales. These are exponential power scales used for raising any number to a power (the range would normally be from about 0.00005 to 20,000).

The LL scales are useful for power calculations that are more complicated than basic squares and cubes, provided by the A, B and K scales. They also allow raising of non-integer numbers to non-

x

integer powers. Another important use of the LL scales is for the calculation of e in hyperbolic functions. The third example of a log scale is the Ln scale.

3 What then is the Ln scale? The Ln scale (the subject of this paper) is a linear scale, similar to the L scale, providing the natural logarithm of numbers from 1 to 10 when used with the C and the D scale. Note that the Ln scales provide natural logarithms in the reverse fashion to the Log Log scales. As x a consequence using the Ln scale to find e will provide the answer on the D scale (or C scale if the Ln scale is on the slide). In either case this is a more convenient arrangement if multiplication x by e is required.

4 Invention of the Ln scale The Ln scale was invented by a high school student, Stephen B. Cohen, in 1958. The original intent was to allow the user to select an exponent x (in the range 0 to 2.3) on the Ln scale and read ex on the C (or D) scale and e–x on the CI (or DI) scale. Pickett and Eckel were given exclusive rights to the scale in the early sixties. Later, Stephen Cohen created a set of "marks" on the Ln scale to extend the range beyond the 2.3 limit, but Pickett never incorporated these marks on any of their slide rules.

4.1 Patent It is probably simpler for me to cite the patent at this point. I cite the words from the Canadian patent CA 671991 (the only patent available to me at the time of writing) which quotes the inventors as Maurice L. Hartung and Stephen B. Cohen (both citizens of the USA) and the owners as Pickett and Eckel. The Canadian patent dates from 1963 whereas I believe the American patent extended from 1958. “This invention relates to the a slide rule having an Ln-L scale arrangement for simplifying the use of a slide rule in computations involving the base e, reducing considerably the number of settings required in comparisons to the usual slide rule when this base is used. The present invention provides a slide rule including the combinations of coextensive logarithmic and linear scales in which for 10 divisions of said logarithmic scale there are 2.302585 + divisions of said linear scale, and a cursor movable along said scales and having a hairline against which they may be read enabling the direct finding of powers of e and logarithms to the base e under the hairline of said cursor. A further object of the invention is to provide an Ln scale which is uniform or linear the same as a L scale, but bears relation to the L scale of the constant log e or 10 or 1 ÷ M = 2.302585 + for the full length of the Ln scale compared to 1.0 for the full length of the L scale whereby the Ln scale may be used in problems involving the base e in like manner as the L scale is used with problems involving the base 10. Another object is to provide an Ln scale which may be combined with out slide rule scales such as C and D and bears relation to the C and D scales of the constant e or Naperian base (2.7182818 +1), and may also be combined with the CI and DI sales to find the powers of e, logarithms to the base, logarithms of proper fractions, powers for negative exponents, and mantissas of logarithms by direct readings cross from the Ln an L scales to the C, D, CI and DI scales.

Another object is to provide a slide rule having Ln, L and C and / or D scales bearing the relationship per scale length in the following ratio: Ln = 2.302585 + L = 1.0 C = 10.0 D = 10.0 and Wherein e of the C and D scales = 2.7182818 + and are coincident with 1 on the Ln scale, the values stated being more accurately identified as; 2.302585092994045684017991454684 + and 2.718281828459045235360287471353 + respectively, According to the MacMillan Logarithmic and Trigonometric Tables, Page 133. Still another object is to provide Ln and L scales of linear or uniform progression used in combination with other scales of logarithmic progression involving relative settings of the body and slide of the slide rule whereby multiplication and division with powers, the logarithms of combined operations, the power of other bases, and hyperbolic functions may be determined on a slide rule of our design. A further object is to provide a slide rule so scaled as to make possible combined operations with the powers of e and logarithms to the base e, with the results accurate to three or four significant figures and readable directly without the necessity of reading a value off one scale and then setting it on another one in order to continue with another calculation. Thus a series of calculations involving the base e may be performed without the use of a log log scale and without the necessity of reading off intermediate values and resetting scales, the final result only being directly read on the appropriate scale. Still a further object is to provide a slide rule having a scale for the base e capable of doing everything that the L scale does for the base 10. An additional object is to provide a scale for a slide rule which makes possible the reading of e to any exponent between 0 and 2.3 directly on the C or D scale s of the rule, or e to any exponent between 0 and 2.3 directly on the CI or DI scales, as well as e to any power between 1 and 10 on the C or D scales, or e to any power between 1 and -0 on the CI or DI scales. Another additional object is to provide an Ln scale so related to all logarithmic scales of a slide rule that problems involving any logarithmic scale and the base w are greatly simplified , and particularly so in combined operations, in comparison with such problems when worked on a slide rule having no Ln scale. With these and other objects in view, our invention consists in the construction, arrangement and combination of the various of our Ln-L scale slide rule, whereby the objects above contemplated are attained, as hereinafter moiré full set forth, pointed out in our claims and illustrated in detail on the accompanying drawings…”

4.2 Instructions for the Use of the Ln scale The instructions for the use of the Ln scale can be studied from numerous Pickett slide rule instruction manuals. The instructions for the use of guide marks however were never published by Pickett and Eckel. To overcome this omission, the instructions have been transcribed and included in the appendices to this paper. There are separate instructions, one for when the Ln scale is on the body of the rule and another for when the Ln scale is on the slide.

4.3 Steve Cohen I do not personally know Stephen Cohen and I have never seen a picture of Stephen. The following is a compilation of a number of emails that may give an insight to the person that invented this scale. Email to the Slide Rule Forum,

“I invented the Ln scale 40+ years ago while still in high school. Some time in the past, I think I sent one of this group's (IRSG) members a not-very-legible copy of instructions for the Ln scale. I thought it was posted as a file, but I can't find it. Perhaps it was too 1 illegible. (Remember, those were the days of carbon paper.) “I also created "gauge marks" to enable quick extension of the scale to ranges beyond 1 to 10 (both above [10-100, etc.]; and below [0.1 1, etc.], but I could not convince Pickett to add the marks to the scale. Pickett did say that it went on the Apollo moon flights as a scale on the Pickett model 600. Anyway, somewhere in the past of this forum there are several messages about the scale. P.S. Please don't ask me for documentation...I'm just too busy. -Steve C”

And another email to the forum, “So much for trying to resurrect 40-yr old memories off the top of one's head. Besides, after working for Uncle Sam for the past 12+ yrs, my brain has softened a bit. (There were, however, a couple of good years.) I have dug through my old papers and found material on the gauge marks. Their design was more ingenious than I remembered. I did use gauge values of 2, 4, 6, etc. as Joe” (Pasquale I believe) “suggests. If I could get interested sufficiently, I'd look closely at Joe's analysis, but my thoughts are elsewhere. ... Anyone interested in working on a logical/statistical structure for dealing with low probability-high consequent events? Anyway, I've found more than one version of the instructions & theory of the marks that I wrote in January, 1960 when I was a college freshman. When I get a chance, I'll decide which one is the latest. If one or more of you wish, I'll then copy & send it to one of you for scanning and you can post it if you wish. Warning: What I have are carbons, so the print is not very clear. I do not have scanner, so can't do it myself. (Still using 9-pin printer, too.) -Steve C

And a further email around the same time, To Joe P, Chris R, etc. The entire reason for the Ln scale was, as Joe correctly knows, to be able to multiply, etc. with x values of e^ = exp(x). Also, as noted, it was limited to the range exp (-2.3) to exp (2.3). I developed "gauge marks" to extend this range significantly, but Pickett would not add them to the scale.

1

(Note: Stephen actually emailed me copies of the carbon copies of the instructions for the usage of gauge marks with the Ln scales. My partner, Dianne has actually transcribed these copies into electronic format so that the slide rule community can have them eternally).

If I recall correctly (we're going back 40+ years now!) you would align a gauge mark on the Ln scale on the slide with an index (1) on the D scale, and then find exp(x) on the D scale. Example: To find exp (222) on the D scale, you would align the 10^2 gauge mark on the Ln scale with the left index on the D scale. You would then set your cursor to 2.22 on the Ln scale and read exp (222) under the hairline on the D scale. I cannot remember if Maurice Hartung wrote the "L-Ln" scale manual or if I ghost wrote it. He was considered a "well known" S-R expert so Pickett wanted to use his name. I do recall ghost writing some manuals while in high school or college, but can't remember which ones. By the way, Ross Pickett once told me that Eckel was the slide rule guru, while Ross was the businessman/marketer. Eckel later had his own company which sold two circular slide rule models. Neat! But apparently sales were poor. What "did-in" Pickett was its move to Santa Barbara, CA. I suspect it borrowed more than it could repay out of its sales and it ended up selling out to Times-Mirror Corp. P.S. I suspect that some of this e-group's members are retired, judging from the number of daily postings. Regrettably I am not. (I never knew why my friends who worked for the Federal Govt. couldn't wait to retire. I now know.) Anyway, the postings are too numerous for me to follow. By the way, I have a Ph.D. in pure math, but everyone thinks it's in statistics. At one time I was so familiar with the Pickett Model 600 that I could picture it in my head and use the "picture" for approximate calculations. Now I can hardly remember my name. Rule on! --Steve C

My lasting impression of Steve, from these emails and others, is of a very busy man who has too many interests and competing requirements in life to reflect on past glories, but continues to seek out new ideas.

5 Uses of the Ln scale As described in the preceding sections, the fundamental reason for the Ln scale was to enable x convenient multiplication of e by another value. This is especially useful in many problems in electrical and electronic engineering. Such as exponential rise or decay illustrated in the following diagrams;

Figure 2

Exponential decay across the resistor in the above circuit is described by the function:

The graph on the right-hand side is described by the following function:

The same equations are similar for resistor capacitor circuits. These equations and variations of them are extremely common throughout electrical and electronic systems. x All would involve some calculation in the form of A  e It is no wonder that the Ln scale appeared on many of the “electronic” [Ref 5] Pickett rules. It is interesting to note that the with the scale directly limited to plus or minus 2.3 decay or rise rates of 2.3 10 seconds ( i.e. e ) it is more suited to electronic circuits than large electric systems, with their longer rise and decay times . Large electric circuits have larger capacitance, smaller inductance and frequency values than electronic circuits and therefore longer rise and decay rates. Apart from the above use, on a slide rule that has only limited real estate for scales, the inclusion of an Ln scale allowed for reasonably quick calculation of hyperbolic functions. Particularly if the Ln scale was on the body and a C and CI scale was included on the slide. With this arrangement x -x both e and e could be read directly and used to calculate the various hyperbolic functions.

6 Extension of the Ln scale As can be seen on many rules with Ln scales, the scale is indexed to either the C or D scales (and conversely with the CI and DI scales). This limited the range from exp (-2.3) to exp (2.3). Thus for values of x > 2.3 or < -2.3 a work around was needed.

6.1 Extending the Range The Ln scale can be extended reasonably easy in a similar fashion to any extension of the L scale. This is demonstrated by the following examples.

6.1.1 Finding the Natural Log of a number The normal method for the extension of the L scale is to express the number in scientific form i.e. a decimal numeral and exponent of 10, for example: 5,790,000 becomes 5.79 X 106 Then setting 5.79 on the C or D scale read the Log10 value on the L scale and add the exponent to this figure, Log10 5.79 + 6 = 0.763 + 6 = 6.76 The method of extension of the Ln scale is remarkably similar, first express the number in scientific terms as above, then find the natural log of the factor 5.79, Ln 5.79 = 1.76 To express the number 106 easily in terms of e, it is convenient to remember that Ln of 10 is approximately equal to 2.3, therefore 106 is 6 times 2.3. Thus the Ln of 5.79 = 1.76 + (6 X 2.3) = 1.76 + 13.8 = 15.56

x

6.1.2 Finding e for -2.3 >X >2.3 A reverse procedure to that above can be used. Firstly divide the exponent x by 2.303 to determine an integral quotient and a remainder. 6.54

For example to find e Divide 6.54 by 2.303 to get 2.84 I.e. a quotient of 2 And the remainder is 6.54 – (2 X 2.303) =1.93 x Find e of 1.93 = 6.89 6.54 x 2.303 is equal to 6.89 X (e2 ) Thus e I.e. 6.89 X 100 = 689 The actual value is 692. Therefore using the above method has resulted in an error of approximately 0.5%. To electronic engineers accustomed to resistor values with tolerances much greater than this, this is an acceptable degree of accuracy.

6.2 Guide Marks Even though the above methods are reasonably easy and straight forward, the active mind of Stephen Cohen devised another method. Previously it was necessary to use the somewhat “arduous” method in section 6.1.2 when working with numbers outside the normal range of the Ln scale. A much simpler system was devised and proposed to be incorporated into the Ln scale. The system involved the use of special graduations called “guide marks”, which where designed as “guide numbers.” These gave the Ln scale an effective range of -16.3 to +16.3 and the C and CI scales a corresponding range of 107 to 10 -7 approximately. The rules for using the guide marks appeared somewhat complicated at first, but the inventor believed that a little time spent in working the examples made the operation so easy that it became second nature.

Figure 3 Location of Gauge Marks

6.2.1 FINDING NATURAL LOGARITHMS OF NUMBERS GREATER THAN 10 •

•

using powers-of-ten, convert your number into the scientific form a x 10r (where 1 < a < 10) multiply r by 2 and set the left index of C scale over this guide number. (If the slide extends too far and the Ln scale is not below a on the C scale, use the next higher guide number (2r + 2), setting the right index of C over it.)

• •

set the hairline over a on C. Read the value on Ln under the hairline add this value to the guide number to obtain the natural logarithm of your number.

For example to find Ln 11,200: • convert 11,200 to 1.12 x 104 • multiply 4 x 2, getting a guide number of 8. Set the left index of C scale over this guide mark • move the hairline to 1.12 on C, and read 1.324 on Ln under the hairline • add: 8 + 1.324 giving Ln 11,200 = 9.324.

6.2.2 FINDING e TO POWER GREATER THAN 2.3 (ey = a, y> 2.3) •

• • • •

subtract from the power y the guide number n which will leave the remainder y1 between 0 and 2 (y –n = y1, 0< y1 <2) if y1 on Ln is to the left of n, set the right index of C over n set the hairline to y1 on Ln. Read the value a1 (1< a1 < 10) on C under the hairline if the left index of C is over n, a = a1 x 10 n/2 if the right index of C is over n, a = a1 x 10 n/2 -1 13.865

For example to find e ; • subtract n=12 from y=13.85 leaving y1 = 1.85 • since 1.85 on Ln is to the right of n=12, set the left index of C over the guide mark 12 • set the hairline to 1.85 on Ln. Read a1 = 1.035 on C under the hairline • since the left index of C is over the guide mark 12, a = 1.035 x 1012/2 = 1.035 x 106 = 1,035,000.

6.2.3 Further instructions for Guide Marks The instructions for guide marks from Steve Cohen included directions for exponents less than 2.3 and variations for when the Ln scale was on the slide or the body. These instructions are reproduced in the appendices. Also included in the appendices is the theory of guide marks and placement of these marks. There is sufficient information in the documents so that anyone can add these to any slide rule containing an Ln scale if they desire.

6.3 Letters to Pickett There are two letters from Steve Cohen to Pickett regarding the guide marks, 1232 N. San Rd. Tuscan, Arizona January 1, 1959

Mr. John W. Pickett President Pickett & Eckel, Inc 1109 S. Fremont Alhambra, Calif.

Dear Mr. Pickett Enclosed are the instructions for using the GUIDE MARKS for extending the range of the Ln scale, on rules that have the Ln scale on the slide, as the model N4. (Since the instructions are different for using rules with Ln on the body than they are for Ln on the slide, I felt that it would be best to have two separate sets of instructions. Presently, I am working on the instructions for the rules with Ln on the slide, the model N1011). The next time the Ln-L Scale Supplement is printed the following correction should be made: p.11 in the square beginning “Rule for a “e …” line 6 beginning with “…y on L.” should be changed to “…y on Ln”. I got the letter from your secretary requesting me to send you my slide rule history. Unless I can con my English teacher into letting me have the original back, I will have to send you a carbon. Since I didn’t stop to correct the carbons when I wrote the paper, it will take me a few days to make all the typographical corrections in it. To date I have received none of the papers from Professor Hartung and Mr. Rusher.

Sincerely,

Stephen B. Cohen

1232 N. San Rd. Tuscan, Arizona January 27, 1960

Mr. John W. Pickett President Pickett & Eckel, Inc 1109 S. Fremont Alhambra, Calif.

Dear Mr. Pickett Here is the set of instructions for using the “Guide Marks” on rules with the Ln scale on the body. I’m sorry it took so long to finish them, but semester finals came up. I have included a sheet with some corrections for the first set of instructions I sent you on the “Guide Marks” for rules with the Ln scale on the slide. You will note that I included nothing on hyperbolic functions, etc. Since such things simply involve finding e to certain powers and then plugging the values obtained into formulas, I felt that it wasn’t worth the extra space to give examples. Explanation of theory has been deliberately been kept to a minimum to conserve space and to reduce confusion resulting from excessive wordage. If necessary, I can give a more complete explanation of the theory. I trust that everything is going well, and that I shall hear from you soon.

Sincerely,

Stephen B. Cohen

7 Slide rules with the Ln scale 7.1 Pickett The following is a list of Pickett rules that have the Ln scale Model Type

Electronic or Math

10 or 5 inch

Ln scale on Body

N16ES

Electronic

10 inch

N515T Electronic

Electronic

10 inch

N531 CREI Electronic

Electronic

10 inch






N535 Electronic

Electronic

10 inch

N931 Electronic

Electronic

10 inch





Ln Scale on Slide

Model Type

Electronic or Math

10 or 5 inch

Ln scale on Body

N1010 SL

Math

10 inch

N1011

Math

10 inch





N3

Math

10 inch

N3P

Math

5 inch

N4

Math

10 inch

N4P

Math

5 inch

N600

Math

5 inch

N1006

Math

5 inch

C18

Electronic

10 inch

X3

Math

10 inch

X4

Math

10 inch

Ln Scale on Slide












Note : Both T and ES versions are included Table 1

Note 1: the only “electronic” rule that Pickett produced that did not have the Ln scale was the N1020. I offer no reasons why this was the case. It is also interesting to note that no circular Pickett rule had a Ln scale. Note 2: The Pickett rules were produced in a number of different variants. For instance those without N in the model number were most likely early versions made with magnesium, those with T in the model number are white, and those with ES in the model number are in “eye saver” yellow. Those with X in the model number are for executive gifts. As a general rule only those rules with an N in the model number could have an Ln scale as those without the N would have been produced before the invention of the Ln scale. Note 3: I believe the optimal layout requires the Ln scale to be on the body of the rule and the slide to have a CI scale on the slide. This arrangement allows chain calculations for both positive and negative powers of e to be easily made. An equivalent arrangement for the Ln scale on the slide would be to have a D and DI scale on the body.

Electronic Type

Figure 4 - from Top to Bottom N515-T, N531-ES, N535-T, N16-ES

Note: due to the absence of labeling of scales it should be noted that the second scale from the top of the 535 is the Ln scale.

Math

Figure 5 - from Top to Bottom N 1010SL-ES, N 1011-T

Figure 6 - from Top to Bottom N3-ES, Pickett N4-ES

Figure 7 -N 600 (the Moon Rule)

Figure 8 – N 1002

7.2 Other Pickett held the patents for the Ln scale until they went out of the business of making slide rules. And it would be easy enough to assume that no other manufacturer made a rule with the Ln scale as Pickett did not (to my knowledge) license the use by others. However, I do have in my possession a non-Pickett Rule that has an Ln scale.

Figure 9 - Westec “Electronics Slide Rule”

Figure 10 - Westec “Electronics Slide Rule”

As can be seen this is a variant of the Pickett 515, Aristo 10175 layout. It is labeled “WESTEC” Electronics Slide Rule and is made for Westechno Ltd, Exmouth, Devon, England. On the reverse side it contains the model number P2361 and bears the BRL trademark. On the front of the rule in the bottom right-hand corner are the words “Prov Pat No. 49828” and “Copyright Reserved”.

8 Conclusion As stated earlier it is no wonder that the Ln scale appeared on many of the “electronic” Pickett rules. The inclusion of an Ln scale allowed for reasonably quick calculation of hyperbolic functions and the powers of e. Although the scale is directly limited to plus or minus 2.3 and therefore more suited to electronic circuits than large electric systems, with their longer rise and decay times it would have been useful in this field of work and I wonder why it wasn’t adopted by more manufacturers. Was it because Pickett vigorously defended their patent (I have no evidence that that was the case) or was it that other manufacturers saw little advantage? In the case of “Elektro” rules all the usual manufacturers continued with two Log Log scales.

9 Acknowledgements I would like to acknowledge the assistance offered by Steve Cohen to me when, a number of years ago, I enquired about the Ln scales. Steve provided all the information that has enabled me to produce this paper without any question. I would also like to thank David Rance whose invaluable skills in dissecting a draft and turning it into a worthwhile paper is without doubt beyond peer.

10 References Ref 1 - Various Pickett slide rule manuals. Ref 2 - ISRG email group archives Ref 3 - JOS Vol. 10 No. 7 2007 Pg 15 Ref 4 - GriffenFly slide rule application Ref 5 - Elektro Rules - Their Scales and Uses, IM2007, R Adams

Appendix A (Ln scale on the body) How to use the

GUIDE MARKS

for EXTENDING the RANGE of the

Ln SCALE (for rules with the Ln scale on the body)

by

Stephen B. Cohen

TABLE OF CONTENTS Introduction ……………………………………………………………….

1

(A)

Finding Natural Logarithms of Numbers Greater than 10 …………..

2

(B)

Finding Natural Logarithms of Numbers Less than 0.1 ……………….

3

(C)

Finding e to Powers Greater than 2.3 ………………………………………… 5

(D)

Finding e to Powers Less than -2.3 …………………………………………

(E)

Computations Involving Powers of e ………………………………………… 8

(F)

Theory ------- How the Guide Marks work ………………………………

6

10

INTRODUCTION

Previously it was necessary to use a rather complication method when working with numbers outside the normal range of the Ln scale. A much simpler system has been devised and incorporated into your scale. This system involves the use of a number of special graduations called “guide marks,” which are designed by “guide numbers.” These give the Ln scale an effective range of -16.3 to +16.3 and the C and CI scales a corresponding range of 107 + to 10 -7 – approx. For numbers in this range, this new method takes the place of the one given in sections (e) – (g) of the manual. (The methods for using common logarithms and power of 10 remain the same.) The rules for using the guide marks may seem somewhat complicated at first, but a little time spent in working the examples will show you that they are actually very simple. A helpful hint : When the left or right index of the C scale is set over a guide mark, this index takes on the value 10 n/2 (where n is the guide number), and when the left or right index of the CI scale is set over a guide mark, that index takes on the value 10 n/2 . All other values on the C and CI scales are then in relation to these values.

(2) (A)

FINDING NATURAL LOGARITHMS OF NUMBERS GREATER THAN 10

RULE : 1)

Using powers-of-ten, convert your number into the form a1 x 10r (where 1 < a1 < 10).

2)

Multiply r by 2 and set the left index of C scale over this guide number. (If the slide extends too far and the Ln scale is not below a1 on the C scale, use the next higher guide number (2r + 2), setting the right index of C over it.)

3)

Set the hairline over a1 on C. Read the value on Ln under the hairline.

4)

Add this value to the guide number to obtain the natural logarithm of your number.

EXAMPLES: 1.

Find Ln 11,200 1)

Convert 11,200 to 1.12 x 104.

2)

Multiply 4 x 2, getting a guide number of 8. Set the left index of C scale over this guide mark.

2.

3)

Move the hairline to 1.12 on C, and read 1.324 on Ln under the hairline.

4)

Add: 8 + 1.324 giving Ln 11,200 = 9.324.

Find Ln 870,000 1)

Convert 870,000 to 8.70 x 105.

2)

Multiply 5 by 2, getting a guide number of 10. When you set the left index of C over this guide mark, you find that the slide extends too far. Therefore, use the next higher guide number, 12 (10 + 2) and set the right index of C over it. (3)

3)

Move the hairline to 8.70 on C. Read 1.676 on Ln under the hairline.

4)

Add: 12 + 1.676 giving Ln 870,000 = 13.676.

PROBLEMS FOR PRACTICE: a)

Ln 670 = 6.570

b)

Ln 34 = 3.526

e)

Ln 850 = 6.745

(B)

FINDING NATURAL LOGARITHMS OF NUMBERS LESS THAN 0.1

RULE : 1)

d)

Ln 455,000 = 13.028 e)

f)

Ln 11,300 – 9.333

Ln 925,000 = 13.738

(consider all values on Ln and guide numbers as positive.)

Using powers-of-ten, convert your number into the form a2 x 10r (Where 0.1
2)

Multiply r by 2 and set the left index of CI over this guide number, (If the slide extends too far, and a2 on the CI scale is not above the Ln scale, use the next higher guide number (2r +2), setting the right index of CI over it.)

3)

Set the hairline to a2 on CI. Head the value on Ln under the hairline.

4)

Add this value to the guide number and attach a minus sign to obtain the natural logarithm of your number.

EXAMPLES: a)

Find Ln 0.000082. 1)

Convert 0.000082 to 0.82 x 10 -4

2)

Multiply -4 by 2, getting a guide number of 8 (remember that

(4) all guide numbers are to be considered positive). Set the left index of CI over the guide mark 8.

b)

3)

Set the hairline to 0.82 on CI and read 1.409 on Ln under the hairline.

4)

Add 8 + 1.409 and attach a minus sign, obtaining Ln 0.000082 = -9.409.

Find Ln 0.00103. 1)

Convert 0.011103 to 0.103 x 10 -2.

2)

Multiply -2 by 2, getting a guide number of 4. When you set the left index of CI over this guide mark, you find that the slide extends too far. Therefore use the next higher guide number, 6 (4 + 2) and set the right index of CI over it.

3)

Move the hairline to 0.103 on CI and read .878 on Ln under the hairline.

4)

Add 6 + .878 and attach a minus sign to obtain Ln 0.00103 = -6.878.

PROBLEMS FOR PRACTICE : a)

Ln 0.070 = -2.659

d)

Ln 0.093 = -2.375

b)

Ln 0.000345 = =7.972

e)

Ln 0.000000622 = -14.290

c)

Ln 0.0000262 = -10.550

f)

Ln 0.00000117 = -13.659

(5) (C)

FINDING e TO POWER GREATER THAN 2.3 ( ey = a, y> 2.3) RULE :

1)

Subtract from the power y the guide number n which will leave the remainder y1 between 0 and 2 (y – n = y1, 0< y1 <2).

2)

a. If y1 on Ln is to the left of n, set the right index of C over n.

3)

Set the hairline to y1 on Ln. Read the value a1 (1< a1 < 10) on C under the hairline.

4)

a. If the left index of C is over n, a = a1 x 10n/2 b. If the right index of C is over n, a = a1 x 10n/2-1

EXAMPLES : a)

Find e 13.865. 1)

Subtract n=12 from y=13.85 leaving y1 = 1.85.

2)

Since 1.85 on Ln is to the right of n=12, set the left index of C over the guide mark 12.

3)

Set the hairline to 1.85 on Ln. Read a1 = 1.035 on C under the hairline.

4)

Since the left index of C is over the guide mark 12, a = 1.035 x 1012/2 = 1.035 x 106 = 1,035,000.

b)

Find e 6.72. 1)

Subtract n=6 from y=6.72 leaving y1 = .72.

2)

Since .72 on Ln is to the left of n=6, set the right index of C over the guide mark 6.

(6) 3)

Set the hairline to .72 on Ln. Read a1 = 8.29 on C under the hairline.

4)

Since the right index of C is over the guide mark 6, A = 8.29 x 106/2-1 = 8.29 x 102 = 829.

PROBLEMS FOR PRACTICE :

(D)

a)

e3.72 = 41.3

d)

e13.46 = 701,000

b)

e7.24 = 1394.

e)

e2.55 = 12.81

c)

e11.30 = 80,000.

f)

e8.11 = 3330

FINDING e TO POWER LESS THAN -2.3 (ey = 1, y < -2.3)

RULE : (Consider all values on Ln, guide numbers, and powers of e as positive). 1)

Subtract from the power y the guide number n which will leave the remainder y1 between 0 and 2 (y –n =y1, 0< y1 <2).

2)

a. If y1 on Ln is to the right of n, set the left index of CI over n. b. If y1 on Ln is to the left of n, set the right index of CI over n.

3)

Set the hairline to y1 on Ln. Read the value a2 (.1 <- a2

<

1.0) on CI under the

hairline. 4)

a. If the left index of CI is over n, a = 12 x 10–(n/2) b. If the right index of CI is over n, a = a2 x 10–(n/2 -1)

EXAMPLES : a)

Find e -7.342 1)

Subtract n=6 from 7.32 (remember that all powers are to be

(7) Considered positive), leaving y1 = 1.32. 2)

Since 1.32 on Ln is to the right of n=6, set the left index of CI over the guide mark 6.

3)

Set the hairline to 1.32 on Ln. Read a2 = 0.662 on CI under the hairline.

4)

Since the left index of CI is over the guide mark 6, a = 0.662 x 10-(6/2) = 0.662 x 103 = 0.000662.

b)

Find e -4.32 1)

Subtract n=4 from y=4.32 leaving y1 = .32.

2)

Since .32 on Ln is to the left of n=4, set the right index of CI over the guide mark 4.

3)

Set the hairline to .32 on Ln. Read a2 = 0.133 on CI under the hairline.

4)

Since the right index of CI is over the guide mark 4, a = 0.133 x 10 –(4/2 – 1) = 0.133 x 10-1 = 0.0133.

PROBLEMS FOR PRACTICE : a)

e -4.62 = 0.00985

d)

e -5.44 = 0.00434

b)

e -7.93 = 0.000360

e)

e -3.30 = 0.0369

c)

e -13.62 = 0.000001216

f)

e -12.77 = 0.00000284

(8)

(E)

COMPUTATIONS INVOLVING POWERS OF e

When working with computations involving multiplications or divisions by powers of e outside the normal range of Ln, the following methods may be used. MULTIPLICATION: (a x ey) 1)

Find the guide number n and the remainder y1 by the rules in sections (C) and (D).

2)

Set the hairline to the guide number n. Pull a on C under the hairline.

3)

Move the hairline to y1 on Ln. Read a x ey on C under the hairline.

4)

If the slide extends too far for a value to be read, set the hairline to the index of C, pull the slide out so that the other index of C is under the hairline, and continue with the problem.

DIVISION : ( a / ey) 1)

Find the guide number n and the remainder y1 by the rules in section (C) and (D).

2)

Set the hairline to y1 on Ln. Pull a on C under the hairline.

3)

Move the hairline to the guide number n. Read (a/ey) under the hairline.

4)

If the slide extends too far for a value to be read, set the hairline over the index of C, pull the other index of C under the hairline, and continue with the problem.

The above rules are for positive powers of e. For negative y remember that multiplication by a negative power is the same as division by a positive power, and vice versa. (9) The above methods may be used with powers of e within the normal range of Ln. Simply use either the right or left index of D instead of the guide number n. EXAMPLES: a)

Find 2.5 x e 6.34

b)

1)

n = 6 and y1 = .34

2)

Set the hairline to the guide number 6. Pull 2.5 on C under the hairline.

3)

Move the hairline to .34 on Ln and read the answer 1415 on C.

Find 72 x e-4.15 1)

n = 4 and y1 = .15

2)

Remember that division with positive powers is the same as multiplication with negative powers). Set the hairline to .15 on Ln. Put 72 on C under the hairline.

3)

Move the hairline to the guide mark 4 and read the answer 1.135 on C

4)

Since the slide extends too far, set the hairline to the right index of C and pull the left index of C under the hairline.

PROBLEMS FOR PRACTICE: a)

14.2 x e8.91

=

105,200

b)

0.799/x3.22

=

20.0

c)

36200 x e-9.14

=

3.88

d)

0.00611/e4.40

=

0.0000750

e)

9.92 x e-6.66

=

0.01272

f)

2.20/e12.21

=

0.00001096 (10)

(F)

THEORY …….. HOW THE GUIDE MARKS WORK

To find the natural logarithm of a number, Ln a, one must first convert it into the form a1 x 10n, 1< a1 < 10. Then, Ln a = Ln a1 + n (Ln 10) = Ln a1 + n (2.3025850930). Since Ln is an evenly divided scale, values may be added or subtracted on it.

A Range including a

B Value of n(Ln10) Added to Ln a1 to Obtain Ln 1

C Guide number

D Position of guide mark on Ln scale

1< a < 101

0.0

0

0.0

101< a < 102

2.30258 50930

2

0.30258 50930

102< a < 103

4.60517 01860

4

0.60517 01860

4

6.90775 52790

6

0.90775 52790

104< a < 105

9.21034 03720

8

1.21034 03720

105< a < 106

11.51292 54650

10

1.51292 54650

106< a < 107

13.81551 05580

12

1.81551 05580

107< a < 107+

16.11809 56310

14

2.11809 56510

10

3<

<

a 10

By setting an index of C scale over the guide mark, the value in column D is automatically added to (or subtracted from) Ln a1. The value in column C is then mentally added (or subtracted) to give Ln a. (Numbers less than 1 are converted into the form a2 x 10n, 0.1 < a2 < 1.0, and the CI scale is used instead of the C scale.) Powers of e are found by the inverse of the above method. ** numerical values were obtained from C.R.C. Standard Mathematical Tables, Eleventh Edition, Cleveland, Ohio Chemical Rubber Publishing Company, 1957 p.173. Corrections to be made in the copy of “How to use the GUIDE MARKS for EXTENDED the RANGE of the Ln SCALE (for rules with the Ln scale on the slide)” By Stephen B. Cohen CONTENTS PAGE: line 6 (E)

Computations Involving ….NOT (E)

Computations Onvolving ….

PAGE 1 : line 7 …. of -16.3 to +16.3 and the D and DI scales a corresponding ….. NOT ….. of -16.3 to _16.3 and the D_scale_ a corresponding …

PAGE 2 : line 7 …. It over the right index of D.) NOT ….. it over the right index of D._ PAGE 3: last line The last word, “the” on this page should be deleted.

Appendix B (Ln scale on the Slide)

How to use the

GUIDE MARKS

for EXTENDING the RANGE of the

Ln SCALE (for rules with the Ln scale on the slide)

by

Stephen B. Cohen

TABLE OF CONTENTS Introduction ………………………………………………………………….

1

(G)

Finding Natural Logarithms of Numbers Greater than 10 ………….. `

2

(H)

Finding Natural Logarithms of Numbers Less than 0.1 ……………….

3

(I)

Finding e to Powers Greater than 2.3 ………………………………………… 5

(J)

Finding e to Powers Less than -2.3 ………………………………

(K)

Computations Involving Powers of e ………………………………………….. 8

(L)

Theory ------- How the Guide Marks work ………………………………….

6

10

INTRODUCTION

Previously it was necessary to use a rather complicated method when working with numbers outside the normal range of the Ln scale. A much simpler system has been devised and incorporated into your scale. This system involves the use of a number of special graduations called “guide marks,” which are designed by “guide numbers.” These give the Ln scale an effective range of -16.3 to +16.3 and the D and DI scales a corresponding range of 107 + to 10 -7 – approx. For numbers in this range, this new method takes the place of the one given in sections (e) – (g) of the manual. (The methods for using common logarithms and power of 10 remain the same.) The rules for using the guide marks may seem somewhat complicated at first, but a little time spent in working the examples will show you that they are actually very simple. A helpful hint: When the left or right index of the D scale is set over a guide mark, this index takes on the value 10n/2 (where n is the guide number), and when the left or right index of the DI scale is set over a guide mark, that index takes on the value 10n/2 . All other values on the D and DI scales are then in relation to these values.

(2) (A)

FINDING NATURAL LOGARITHMS OF NUMBERS GREATER THAN 10 RULE:

5)

Using powers-of-ten, convert your number into the form a1 x 10r (where 1 < a1 < 10).

6)

Multiply r by 2 and set the left index of D scale over this guide number. (If the slide extends too far and the Ln scale is not below a1 on the D scale, use the next higher guide number (2r + 2), setting the right index of D).

7)

Set the hairline over a1 on D. Read the value on Ln under the hairline.

8)

Add this value to the guide number to obtain the natural logarithm of your number.

EXAMPLES: 1.

Find Ln 11,200 5)

Convert 11,200 to 1.12 x 104.

6)

Multiply 4 x 2, getting a guide number of 8. Set the guide mark 8 over the left index of D scale.

2.

7)

Move the hairline to 1.12 on D, and read 1.324 on Ln under the hairline.

8)

Add: 8 + 1.324 giving Ln 11,200 = 9.324.

Find Ln 870,000 5)

Convert 870,000 to 8.70 x 105.

6)

Multiply 5 by 2, getting a guide number of 10. When you set this guide mark over the left index of D, you find that the slide extends too far. Therefore, use the next higher guide number, 12 (10 + 2) and set it over the right index of D.

(3) 7)

Move the hairline to 8.70 on D. Read 1.676 on Ln under the hairline.

8)

Add: 12 + 1.676 giving Ln 870,000 = 13.676.

PROBLEMS FOR PRACTICE: a)

Ln 670 = 6.570

d)

Ln 455,000 = 13.028

b)

Ln 34 = 3.526

e)

Ln 850 = 6.745

(B)

FINDING NATURAL LOGARITHMS OF NUMBERS LESS THAN 0.1 RULE:

e) f)

Ln 11,300 – 9.333

Ln 925,000 = 13.738

(consider all values on Ln and guide numbers as positive.) 5)

Using powers-of-ten, convert your number into the form a2 x 10r (Where 0.1 < a2

6)

<

1.0).

Multiply r by 2 and set this guide number over the left index of D. (If the slide extends too far, and the Ln scale is not above a2 on the DI scale, use the next higher guide number (2r +2), setting the right index of D)

7)

Set the hairline to a2 on DI, (you probably will have to turn the rule over). Read the value on Ln under the hairline.

8)

Add this value to the guide number and attach a minus sign to obtain the natural logarithm of your number.

EXAMPLES: a)

Find Ln 0.000082. 1)

Convert 0.000082 to 0.82 x 10 -4

2)

Multiply -4 by 2, getting a guide number of 8 (remember that

(4) all guide numbers are to be considered positive). Set the guide mark 8 over the left index D.

b)

3)

Set the hairline to 0.82 on DI and read 1.409 on Ln under the hairline.

4)

Add 8 + 1.409 and attach a minus sign, obtaining Ln 0.000082 = -9.409.

Find Ln 0.00103. 1)

Convert 0.011103 to 0.103 x 10 -2.

2)

Multiply -2 by 2, getting a guide number of 4. When you set this guide mark over the left index of D, you find that the slide extends too far. Therefore use the next higher guide number, 6 (4 + 2) and set it over the right index of D.

3)

Move the hairline to 0.103 on DI and read .878 on Ln under the hairline.

4)

Add 6 + .878 and attach a minus sign to obtain Ln 0.00103 = -6.878.

PROBLEMS FOR PRACTICE: a)

Ln 0.070 = -2.659

d)

Ln 0.093 = -2.375

b)

Ln 0.000345 = =7.972

e)

Ln 0.000000622 = -14.290

c)

Ln 0.0000262 = -10.550

f)

Ln 0.00000117 = -13.659

(5) (C)

FINDING e TO POWER GREATER THAN 2.3 ( ey = a, y> 2.3) RULE :

1)

Subtract from the power y the guide number n which will leave the remainder y1 between 0 and 2 (y –n = y1, 0< y1 <2).

2)

a. If y1 on Ln is to the right of n, set n over the left index of D.

3)

Set the hairline to y1 on Ln. Read the value a1 (1< a1 < 10) on C under the hairline.

4)

a. If n is over the left index of D, a = a1 x 10n/2 b. If n is over the right index of D, a = a1 x 10n/2-1

EXAMPLES: a)

Find e 13.865. 1)

Subtract n=12 from y=13.85 leaving y1 = 1.85.

2)

Since 1.85 on Ln is to the right of n=12, set the guide rule mark as over the left index of D.

3)

Set the hairline to 1.85 on Ln. Read a1 = 1.035 on D under the hairline.

4)

Since the guide mark 12 is over the left index of D, a = 1.035 x 1012/2 = 1.035 x 106 = 1,035,000.

b)

Find e 6.72. 1)

Subtract n=6 from y=6.72 leaving y1 = .72.

2)

Since .72 on Ln is to the left of n=6, set the guide mark 6 over the right index of D.

(6) 3)

Set the hairline to .72 on Ln. Read a1 = 8.29 on D under the hairline.

4)

Since the guide mark 6 is over the right index of D, a = 8.29 x 106/2-1 = 8.29 x 102 = 829.

PROBLEMS FOR PRACTICE:

(D)

a)

e3.72 = 41.3

d)

e13.46 = 701,000

b)

e7.24 = 1394.

e)

e2.55 = 12.81

c)

e11.30 = 80,000.

f)

e8.11 = 3330

FINDING e TO POWER LESS THAN -2.3 (ey = 1, y < -2.3)

RULE: (Consider all values on Ln, guide numbers, and powers of e as positive). 1)

Subtract from the power y the guide number n which will leave the remainder y1 between 0 and 2 (y –n =y1, 0< y1 <2).

2)

a. If y1 on Ln is to the right of n, set n over the left index of D. b. If y1 on Ln is to the left of n, set n over the right index of D.

3)

Set the hairline to y1 on Ln. Read the value a2 (.1
4)

a. If n is over the left index of D, a = 12 x 10 –(n/2) b. If n is over the right index of D, a = a2 x 10 –(n/2 -1)

EXAMPLES: a)

Find e -7.342 1)

Subtract n=6 y 7.32 (remember that all powers are to be

(7) Considered positive), leaving y1 = 1.32. 2)

Since 1.32 on Ln is to the right of n=6, set the guide mark 6 over the left index of D.

3)

Set the hairline to 1.32 on Ln. Read a2 = 0.662 on DI under the hairline.

4)

Since the guide mark 6 is over the left index of D, a = 0.662 x 10-(6/2) = 0.662 x 103 = 0.000662.

b)

Find e -4.32 1)

Subtract n=4 from y=4.32 leaving y1 = .32.

2)

Since .32 on Ln is to the lef6t of n=4, set the guide mark 4 over the right index of D.

3)

Set the hairline to .32 on Ln. Read a2 = 0.133 on DI under the hairline.

4)

Since the guide mark 4 is over the right index of D, a = 0.133 x 10 –(4/2 – 1) = 0.133 x 10-1 = 0.0133.

PROBLEMS FOR PRACTICE: a)

e -4.62 = 0.00985

d)

e -5.44 = 0.00434

b)

e -7.93 = 0.000360

e)

e -3.30 = 0.0369

c)

e -13.62 = 0.000001216

f)

e -12.77 = 0.00000284

(8)

(E)

COMPUTATIONS INVOLVING POWERS OF e

General rules multiplications and divisions involving ay 1)

Positive powers (y > 0): Use Ln as if it were a C scale.

2)

Negative powers (y < 0): Use Ln as it were a CI scale. (Multiplication by a negative power is the same as division by a positive power and vice versa). (These same rules can be used for 10y and the L scale).

For power outside the normal range the following method may be used: 1)

Find the guide number n and the remainder y1 by the rules in sections (C) and (D).

2)

Using the general rules with y1 in the calculations, consider the guide mark n as the index instead of the 1’s of the C and Ci1 scales.

4)

If the slide extends too far for a value to be read, set the hairline to the 1 of C, pull out the slide so that the other 1 of C is under the hairline, and continue with the problem.

(In involved calculations, always work with the power of e first to avoid making errors). EXAMPLES: a)

Find 2.5 x e 6.34 1)

n = 6 and y1 = .34

2)

Set the hairline to 2.5 on D. Pull the guide mark 6 under the hairline. Move the hairline to .34 on Ln and read the answer 1415 on D.

(9)

b)

Find 72 x e-4.15

1)

n = 4 and y1 = .15

2)

Set the hairline to 72 on D. Pull .15 on Ln under the hairline.

4)

Since the slide extends too far, move the hairline to the left 1 of C, and pull the slide out so that the right 1 of C is under the hairline. Now move the hairline to the guide mark 4 and read the answer 1.135 on D.

PROBLEMS FOR PRACTICE : a)

14.2 x e8.91

=

105,200

b)

0.799/x3.22

=

20.0

c)

36200 x e-9.14

=

3.88

d)

0.00611/e4.40

=

0.0000750

e)

9.92 x e-6.66

=

0.01272

f)

2.20/e12.21

=

0.00001096

(10) (F)

THEORY …….. HOW THE GUIDE MARKS WORK

To find the natural logarithm of a number, Ln a, one must first convert it into the form a1 x 10n, 1 < a1 < 10. Then, Ln a = Ln a1 + n (Ln 10) = Ln a1 + n (2.3025850930). Since Ln is an evenly divided scale, values may be added or subtracted on it. A

B

C

D

Range including a

Value of n(Ln10) Added to Ln a1 to Obtain Ln 1

Guide number

Position of guide mark on Ln scale

1< a < 101

0.0

0

0.0

101< a < 102

2.30258 50930

2

0.30258 50930

102< a < 103

4.60517 01860

4

0.60517 01860

4

6.90775 52790

6

0.90775 52790

104< a < 105

9.21034 03720

8

1.21034 03720

105< a < 106

11.51292 54650

10

1.51292 54650

106< a < 107

13.81551 05580

12

1.81551 05580

107< a < 107+

16.11809 56310

14

2.11809 56510

10

3<

<

a 10

By setting the guide mark over an index of D scale over the guide mark, the value in column D is automatically added to (or subtracted from) Ln a1. The value in column C is them mentally added (or subtracted) to give Ln a. (Numbers less than 1 are converted into the form a2 x 10n, 0.1 < a2 < 1.0, and the DI scale is used instead of the D scale.) Powers of e are found by the inverse of the above method. ** numerical values were obtained from C.R.C. Standard Mathematical Tables, Eleventh Edition, Cleveland, Ohio Chemical Rubber Publishing Company, 1957 p.173. Corrections to be made in the copy of “How to use the GUIDE MARKS for EXTENDED the RANGE of the Ln SCALE

(for rules with the Ln scale on the slide)” By Stephen B. Cohen CONTENTS PAGE : (E)

line 6

Computations Involving ….NOT (E)

Computations Onvolving ….

PAGE 1 : line 7 …. of -16.3 to +16.3 and the D and DI scales a corresponding ….. NOT ….. of -16.3 to _16.3 and the D_scale_ a corresponding … PAGE 2 : line 7 …. It over the right index of D.) NOT ….. it over the right index of D._ PAGE 3: last line The last word, “the” on this page should be deleted.