Slide Rule Super Powers

The Log-Log scales and exponentiation

Napier’s search for a set of logarithms that could be used to turn multiplication and division problems into addition and subtraction operations led him and his colleagues to a Base 10 system, using his “decimal point” for keeping track of factors of 10. Years later it was found that one can think of the logarithm as an exponent applied to the same base number — that numbers could not only be raised to integer powers, but non-integer powers as well. The common logarithm, 𝜆, of a number x is then related to x by

\(x = 10^\lambda\)

from which 𝜆 = log x. But suppose we want to take an arbitrary number a and raise it to an arbitrary power b, that is we want to calculate

\(y=a^b.\)

From the general laws and relationships of logarithms we found that

\(\log y=\log a^b = b\times \log a.\)

And we have used this relationship to find the value of a number raised to a power by using the “Log” scale — labeled L — on a standard slide rule. And the A,B, and K scales are based on this same principle as well.

In 1814 Peter Mark Roget, M.D. and overall renaissance man — yes, the same Roget of the famous “Roget’s Thesaurus” — became the first to note that if a second logarithm was taken of the above result, then the log of the log of y would amount to

\(\begin{eqnarray*} \log \log a^b &=& \log b + \log \log a. \end{eqnarray*}\)

Roget made the proposal that a slide rule with a “log-log” scale provided on its stock, along with the standard “log” scale on its slide, could enable the calculation of a number raised to an arbitrary power using a single setting of the slide. While this development brought Roget recognition, slide rules with these scales remained just a curiosity among the mathematical elite for nearly 90 years. But between 1900 and 1909 the makers Dennert and Pape in Germany and Keuffel and Esser in the United States began manufacturing slide rules with Log-Log scales for use in the engineering and scientific efforts of the day, originally in electrical calculations with such scales found on the early electro rules in Europe.

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Creating a Log-Log Scale

The standard C/D scales on a slide rule are marked with numbers, say b, in the range from 1 ≤ b ≤ 10 in logarithmic fashion, where the distance from 1 is proportional to the common logarithm of the number, log b. The C/D scales go from 1 to 10 since log 1 = 0 and log 10 = 1. We want to create a standard “Log-Log” scale that also has a range of values of 0 to 1, so that we can add “log b” to “log log a” in our calculation, as in our expression above. Since

\(\begin{eqnarray*} 0 &=& \log 1 &=& \log (\log 10) \\ 1 &=& \log 10 &=&~~ \log (\log 10^{10}) \end{eqnarray*}\)

then an appropriate range for our Log-Log scale could be from 10 on the left to 10,000,000,000 on the right. Suppose the C scale has a length of 10 inches from 1 to 10 on our slide rule. We would start our Log-Log scale with a value of 10. Then the distance from the left index over to a general number a will be equal to (10 inches)⋅log log a, and the scale will end at 10 billion. To raise numbers to powers, we use the C scale on the slide to “add” values of log b to values of log log a found on the stock of the slide rule.

For example, below are images taken of a Pickett Model 4 slide rule. The back side of the rule has a scale labeled LL4 which fulfills the requirements just outlined. The images show the beginning and end of the LL4 scale, which satisfy the range in question. The images also show the C/D scales which run from 1 to 10

Left side of LL4 scale at bottom, that starts at 10. At the top of the image is the C scale that starts at 1.

Right side of LL4 scale at bottom, that ends at 10 billion. At the top of the image is the C scale that ends at 10.

As an example, let’s use this scale to find the value for 12 raised to the power 2.7. In the image below, we see that the index of the C scale is aligned with 12 on the LL4 scale. Moving the cursor to 2.7 on the C scale (bottom image), we can read the result under the cursor on the LL4 scale: 820. (By computer: 819.954)

C index set to the 12 on the LL4 scale.

With the cursor set to 2.7 on the C scale, the result on LL4 is about 820.

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Mixed-Base Log-Log Scales

The Pickett example above has log-log scales solely based on the common logarithm. But when log-log slide rules began to appear around 1900, a mix of Base 10 and other bases was employed. The choice was mostly driven through the desire to have a more meaningful range for the Log-Log scale across the (typically) 10-inch slide rule. If we have Base 10 scales such as A and B, or C and D, already on the slide rule, then the Log-Log scales could be a mixture of Base 10 and any other base, since we can write

\(\begin{eqnarray*} y = a^b~~~ \longrightarrow~~~ \log_c y&=&\log_c a^b = b\times\log_c a \\ ~\\ \log \log_c a^b &=& \log b + \log\log_c a. \end{eqnarray*}\)

for any base c. (Note that “log” by itself, without the subscript c, refers to finding the Base 10 logarithm.) Slide rules from the early 1900s are known to have been made using bases of c = 2, 10, and one with a base near 3 made by Faber.³⁵

However, it was realized very quickly that using a mix of common and natural logarithms can be used, as this mix also has other advantages. This time, to compute a raised to the power b, we can first find the natural log of both sides, where the base of the natural logarithm is e = 2.718…, and then take the common log of the results:

\(\begin{eqnarray*} y = a^b~~~ \longrightarrow~~~ \ln y&=&\ln a^b = b\times\ln a \\ ~\\ \log \ln a^b &=& \log b + \log \ln a. \end{eqnarray*}\)

Again the C scale can be used to add “log b” to the value of “log ln a” to arrive at the value of “log ln y”, and hence deduce y.

The range across such a Log-Log scale can be found as follows. Here, since

\(\begin{eqnarray*} 0 &=& \log 1 &=& \log (\ln e^1) \\ 1 &=& \log 10 &=&~~ \log (\ln e^{10}) \end{eqnarray*}\)

then this scale should start at e = 2.718, and would have a maximum extent of e raised to the 10th power, which is about 22,000 — much more reasonable, perhaps, than an upper limit of 10 billion. Though this is a much shorter range than for the Base 10 log-log scales, it should cover most of the needs of everyday problems, and one should expect more accuracy in the results for slide rules of the same physical length. In either case, whether base 10 log-log scales are used or mixed base log-ln scales are used, these are all referred to as “Log-Log” scales. They are most frequently labeled “LL” (LL1, LL2, etc.), though other labels are often found such as “Ln”, “ln”, etc. Some of the early Pickett models named their scales N1, N2, etc., for the ones that were totally Base-10 systems.

It was in 1909 when Keuffel and Esser of New York introduced their Model 4092 Log-Log slide rule with e = 2.71828 … as the base c for its Log-Log scales. This quickly became the standard in the industry, due to the usefulness of having the ability to incorporate natural logarithms and exponentials into calculations for finding exponential growth and decay rates, and so on. However, some slide rule makers — Pickett in particular, who began making slide rules in the 1940s — used c = 10 on a few of their models even through the 1970s.

Segmented Log-Log Scales

The two examples of Log-Log (LL) scales discussed above went from 10 to 10 billion (Base 10 only) or from 2.718 to 22,000 (mixed Base e and Base 10), associated with C/D scales that go from 1 to 10. But, what if we want to take a number less than 10 (or less than 2.718) to a power? If we re-interpret the range of the C/D scales, then we can generate separate LL scales for other ranges. That is, for example, if we suppose that C/D represents the range from 0.1 to 1, then we can create a new mixed-base scale that covers this range. It would range from e to the power 0.1 (= 1.1052) to e to the power 1 (= 2.718). The typical modern Log-Log slide rule will contain 3 or 4 scales, which have the limits and labels shown in the following table:

\(\begin{matrix} {\rm {\bf Scale}} & & {\rm {\bf left~limit}} & {\rm {\bf right~limit}} \\ D & x & 1 & 10 \\ LL0 & e^{x/1000} & 1.001 & 1.0101 \\ LL1 & e^{x/100} & 1.0101 & 1.1052 \\ LL2 & e^{x/10} & 1.1052 & 2.7183 \\ LL3 & e^x & 2.7183 & 22026 \\ \end{matrix}\)

Many Log-Log slide rules also have scales for negative exponents, the limits and labels for which are shown here:

\(\begin{matrix} {\rm {\bf Scale}} & & {\rm {\bf left~limit}} & {\rm {\bf right~limit}} \\ D & x & 1 & 10 \\ LL00 & e^{-x/1000} & 0.999 & 0.99 \\ LL01 & e^{-x/100} & 0.99 & 0.9048 \\ LL02 & e^{-x/10} & 0.9048 & 0.3679 \\ LL03 & e^{-x} & 0.3679 & 5\times 10^{-5} \\ \end{matrix}\)

In reality, the four scales LL0-3 can be thought of a single scale that is associated with 4-decade-long C/D scales that run from 0.001 to 0.01 to 0.1 to 1 to 10. (Similarly for LL00-03 scales.) So, suppose we want to compute 1.02 raised to the 35th power. We would find 1.02 on the LL1 mixed-base scale and align the left index of the slide to this number, using the cursor. We would then slide the cursor over to “3.5” on the C scale. The result under the cursor on the LL1 scale would be 1.02 raised to the 3.5 power (1.0717). But, looking at the result on the LL2 scale will be 1.02 raised to the 35th power (2.000). And if you need it, 1.02 raised to the 350th power would be found on LL3 (about 1020). This can be seen directly in our new image below.

Use of Log-Log scales on a K&E Deci-Lon slide rule (virtual) to compute 1.02 raised to the 35th power (and 3.5th and 350th). Virtual rule courtesy of ISRM.

Another strong benefit for using the mixed-base log-ln scale system is that (a) values of the exponential function can be found directly from these scales and (b) the natural logarithm — the inverse of the exponential function — can also be found for a large range of arguments. So for example, if we need exp(2.8), we can put the cursor on 2.8 on D and directly below on the LL3 scale we find a value of about 16.4. And exp(0.28) would be 1.323, found on the LL2 scale. The tables above can help to determine which scale to use for which range of exponential arguments. Working backward, we note that if we want to know the natural logarithm of a number, say of 38, then we can find the number on the LL scales (38 is on the LL3 scale) and look directly at the D scale to read its natural logarithm (ln 38 = 3.638).

It should be noted that on some of the older slide rules the A/B scales might have been used instead of the C/D scales in conjunction with the Log-Log scales. To be certain which set is used, one can easily take a number on the LL scale, find the distance to its square on the same LL scale, and then see whether the A scale or the D scale has this same physical distance between its “1” and “2” marks, for example. Also, just to confuse things further, some slide rules tied the A/B scales to the LL0 (and LL00) scale, and then the C/D scales were used with the LL1-3 scales. This is why the LL0/LL00 scales might be found at the top of the slide rule, and the LL1,2, and 3 scales at the bottom. Even the early version of the Pickett Model 500 from the 1940-50s used such a scheme. Oh, and the original “LL0” scale on the new K&E 4092-3 in the 1920s actually used a split LL0/LL00 scale, all on one line, which took special care to interpret. One always should study the scale arrangement of a particular rule to understand its use, or refer to its instruction book if lucky enough to find a copy.

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The best way to get used to using the Log-Log scales is to use the Log-Log scales. Below are a few examples, which can also be found on my web site. Also found at this link are some suggested solutions. Happy Computing!

Problems:

1. Draw a graph of 1.15 raised to the x over the range of 0.25 < x < 8.

2. The mean lifetime of Carbon-14 is 8267 years. The ratio of 14C/12C in a sample is measured and found to be smaller than today’s current ratio in the air by a factor of 8.4. Estimate the age of the sample.

3. Determine the value of sinh(1.73) and cosh(1.73). Note that

   \(\sinh x = \frac{e^x - e^{-x}}{2}, ~~~~~ \cosh x = \frac{e^x + e^{-x}}{2}.\)

4. An account starting with $10,000 earns 5.7% annual interest over a period of 38 years. Find the final value of the account.

5. Find 0.95 raised to the power 87.

6. The value of the following function of x can be determined through the use of a Taylor Series expansion:

   \(f(x) = \frac{x^2}{4-x^2} = \frac{x^2}{4} + \frac{x^4}{4^2} + \frac{x^6}{4^3} + \ldots, ~~~~~~|x|<2.\)

   Find f(1.05).

7. Determine the natural logarithm of 11,458.

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