Natural Logs with the LL Scales

Okay, we're really going to up the amperage now by switching over to a more endowed engineering type slide rule that'll make short work of natural logs and exponentials. You'll need a rule featuring the so-called LL (log-log) scales.  I'll be using a Keuffel & Esser (K+E) 4181-1 pocket slide rule, simply because I think it's cute and easy on the eyes. But there are many other units at your disposal.

You can actually perform a plentitude of surprising operations with the LL scales, but the best way to master the broad range of what's possible is by first focusing simply on how to use them to compute natural logarithms. In that way, you'll become comfortable with the various intervals at the outset and learn exactly which scale to reach for and when.

Overview of the LL Scales


Recall that the natural log function, ln(x), has domain of all positive reals, and range of all reals. Moreover, the domain splits into two distinct portions: logs of arguments less than 1 are negative, while those greater than 1 are positive.

One other thing: the values of the log function (even though it grows without bound), bunch up as the argument increases. Thus, slide rule manufacturers have seen fit to apportion various subranges over several scales rather than try to cram everything into a single length. On the K+E I'm using which is about 6" long, there are LL1, LL2, LL3, LL01, LL02 and LL03 scales. Juxtapose them all and you've got the equivalent of a 36" slide rule!

Now the most important to learn right away is that the argument, x, is always pinpointed on one of the LL scales, and the result, ln(x) will always be found on the D scale. We recall that D spans 1.0 to 10.0, however when used to output natural logs, we interpret the order of magnitude, so it might mean 0.1 to 1.0, or -10.0 to -1, etc. You'll find this is all quite easy to keep in mind. Furthermore, on my K+E, the manufacturer has even etched the various ranges on the right side of the scales. Nothing to memorize! Refer to the following figure:

Click to enlarge.
Let's begin on the right and work backwards. You'll note that arguments for ln(x) span about 2.718 to 20,000 on the LL3 scale. In this portion of the domain, the range covers 1.0 to 10.0 which is measured on the D scale as always. You no doubt recognize 2.718 as the slide rule's approximation to e, and clearly ln(e) = 1.0.

Move back a notch. If the argument lies between 1.105 and 2.718, now on the LL2 scale, then the associated functional values span 0.1 to 1.0. Again, these are found on the D scale but mentally interpreted to be one order of magnitude less than before. That's not hard to remember, but it's nice that K+E printed that range as a reminder next to the LL2 scale.

Drop lower again. If the argument lies between 1.01 and 1.105 on the LL1 scale, then the associated functional values span 0.01 to 0.1 on the D scale. Are you starting to get comfortable with the pattern?

Now when the argument gets really close to 1.0 on either side, then the logarithm becomes exceedingly tiny and close to 0. So, we pass over that dead region of 0.99 to 1.01. The idea here is that for most engineering application, anytime you're computing the natural logarithm of a number between 0.99 and 1.01, that result is so near 0, it becomes negligible. However, for the fastidious, hang on a few moments, and we'll see some slide rules feature an LL0 scale really allowing you to split hairs.

Continuing our leftward migration in the figure above, the next subrange is 0.905 to 0.99 on the LL01 scale, and it maps to -0.1 to -0.01 on the D scale, after a suitable mental translation for the order of magnitude.

Similarly, arguments between 0.0368 to 0.905 on LL02 go to -1.0 to -0.1 on D.

Finally, 0.0000454 to 0.0368 on the LL03 scale map to -10.0 to -1.0 on the D scale.

In short, we can directly find natural logarithms of arguments from 0.0000454 all the way to 20,000, with a brief timeout when very close to 1. I urge you to study the figure once again and look for how LL03+LL02+LL01 along with LL1+LL2+LL3 are in effect gigantic scales.

With regard to arguments near 1.0, in my collection the Post Versalog 1460 and Faber-Castell 2/83N both sport LL0 and LL00 scales that let you zero in even closer on ln(1) = 0.

A Recap and a Few Extra Details


These are really easy to use, so don't let my long-winded explanation put you off. Nonetheless, a brief summary might be useful just now. And I'll mention a couple other things I passed over above for simplicity.

When computing natural logarithms with the LL scales:
  1. The argument is always specified on an LL scale.
  2. The value is always found on the D scale, with the order of magnitude mentally set.
  3. Most slide rules inidicate the associated order of magnitude next to each LL scale.
  4. The LL1, LL2 and LL3 scales are for arguments greater than 1.
  5. The LL01, LL02 and LL03 scales are for arguments between 0 and 1. They read right-to-left and are typically printed in red. Moreover, the value of ln(x) on any of these three scales is negative.
With that, let's work some problems!

Exercise 1.1: Compute ln(13.7).

The argument is greater than 2.718 (e), so we must be using LL3.
  1. Set the cursor to 13.7 on the LL3 scale.
  2. Read the result under the hairline on the D scale: 2.62. No mental adjustment for order of magnitude is required.
Here's how it looks on my K+E 4181-1:

Cursor to LL3:13.7, result at D:2.62

Exercise 1.2: Compute ln(225).
  1. Set the cursor at 225 on the LL3 scale.
  2. Read the result under the hairline on the D scale: 5.42.
Pictorially:

Cursor to LL3:225, result at D:5.42

Exercise 1.3: Compute ln(1570).
  1. Set the cursor at 1570 on the LL3 scale.
  2. Read the result under the hairline on the D scale: 7.36.
And this is what it looks like:

Cursor to LL3:1570, result at D:7.36

Though covering a fairly broad range of arguments, all three of these problems only required LL3 and D. Let's get used to the other scales now.

Exercise 2.1: Compute ln(2.5).

Aha! Something less than e, so we drop down to the LL2 scale.
  1. Set the cursor at 2.5 on the LL2 scale.
  2. Read the result under the hairline on the D scale, and interpret one order of magnitude less: 0.92.
In summary:

Cursor to LL2:2.5, result at D:0.92

Exercise 3.1: Compute ln(1.05).

We're creeping close to 1 now. Time to go to the LL1 scale.
  1. Set the cursor at 1.05 on the LL1 scale.
  2. Read the result under the hairline on the D scale and mentally adjust for two orders of magnitude smaller: 0.0488.
Graphically:

Cursor to LL1:1.05, result at D:0.0488

We need to see some problems in which the logarithm drops below the x-axis.

Exercise 4.1: Compute ln(0.94).

The argument is below but reasonably close to 1, so we'll require the LL01 scale.
  1. Set the cursor at 0.94 on the LL01 scale.
  2. Read the result under the hairline on the D scale, mentally adjust for the order of magnitude, and prefix with a minus sign: -0.062.
Here's what I see:

Cursor to LL01:0.94, result at D:-0.062

Exercise 4.2: Compute ln(0.52).
  1. Set the cursor at 0.52 on the LL02 scale.
  2. Read the result under the hairline on the D scale, interpret the order of magnitude and negate: -0.654.
In pictures:

Cursor to LL02:0.52, result at D:-0.654

Exercise 4.3: Compute ln(0.17).
  1. Set the cursor at 0.17 on the LL03 scale.
  2. Read the result under the hairline on the D scale, interpret the order of magnitude and negate: -1.77.
Here's how it goes:


Cursor to LL03:0.17, result at D:-1.77

And that's a wrap! I don't know if you suffered through how we calculated natural logs with the L and Ln scales, but if so, you'll appreciate that our new approach with the LL scales is a breath of fresh air. It's pretty much one-stop shopping, especially if the manufacturer has labeled the scales with the various ranges.

Next installment: Powers of e with the LL Scales