Powers of e with the LL Scales

In the last installment, we learned how to compute natural logarithms by means of the LL (log-log) scales. It's a simple matter to turn that method around and come up with e raised to any power. I urge you to review that episode, especially the figure illustrating the the domain and range of ln(x) and how they're apportioned over the LL and D scales.

The rules for finding e^x are quite simple:
  • The argument is always indicated on the D scale.
  • The D scale may represent 1.0 through 10.0, 0.1 through 1.0, -0.01 through -0.1, etc., depending on the argument.
  • The resulting value is always found on one of the LL scales (LL1, LL2, LL3, LL01, LL02 or LL03).
  • The scale to use is determined by the size of the argument.

You might think it would be tricky to know how to interpret the number on D or how to determine which LL scale to use, but it's not. Most slide rules print helpful reminders right on the rule. Here's how my K+E 4181-1 does it:

Click any photo to enlarge

Here, at the bottom, we see that the LL1 scale is meant to be used with arguments ranging from 0.01 to 0.1 (as indicated on the D scale by 1.0 to 10.0).

At the top, we further note that arguments spanning -0.01 to -0.1 are associated with values on the LL01 scale.

On the flip-side of this duplex rule, we find:


LL2 (at the bottom) corresponds to arguments in the 0.1 to 1.0 range, while LL3 (second from the bottom) goes with 1.0 to 10.0. Likewise, LL02 (top) is for arguments from -1.0 to -0.1, and LL03 (second from top) for -10.0 to -1.0.

And remember, the LL scales for negative arguments read right-to-left. With that, let's tuck into some exercises.

Positive Arguments


Exercise 1.1: Compute e^3.7.
  1. Set the cursor to 3.7 on the D scale.
  2. Read the result under the hairline on the LL3 scale: 40.5.
In summary:

Cursor to D:3.7, result at LL3:40.5

Exercise 1.2: Compute e^7.
  1. Set the cursor to 7 on the D scale.
  2. Read the result under the hairline on the LL3 scale: 1100.
Here's what it looks like:

Cursor to D:7, result at LL3:1100

In the next problem, we switch to the LL2 scale because the argument lies between 0.1 and 1.0, as the label on the slide rule reminds us.

Exercise 2.1: Compute e^0.19.
  1. Set the cursor to 0.19 (1.9) on the D scale.
  2. Read the result under the hairline on the LL2 scale: 1.21.
Graphically, we have:

Cursor to D:0.19, result at LL2:1.21

Are you getting the hang of it? The next problem drops another order of magnitude lower.

Exercise 3.1: Compute e^0.065.
  1. Set the cursor to 0.065 (6.5) on the D scale.
  2. Read the result under the hairline on the LL1 scale: 1.0672.
In pictures:

Cursor to D:0.065, result at LL1:1.0672

Negative Arguments


The reverse-reading LL01, LL02 and LL03 scales are employed when the argument is less than zero.

Exercise 4.1: Compute e^-0.025.
  1. Set the cursor to -0.025 (2.5) on the D scale.
  2. Read the result under the hairline on the LL01 scale: 0.9753.
Summarizing the steps:

Cursor to D:-0.025, result at LL01:0.9753

Exercise 4.2: Compute e^-0.42.
  1. Set the cursor to -0.42 (4.2) on the D scale.
  2. Read the result under the hairline on the LL02 scale: 0.657.
Pictorially:

Cursor to D:-0.42, result at LL02:0.657

Exercise 4.3: Compute e^-3.9.
  1. Set the cursor to -3.9 (3.9) on the D scale.
  2. Read the result under the hairline on the LL03 scale: 0.02.
And it looks like this:

Cursor to D:-3.9, result at LL03:0.02

You should have had no trouble determining which scale to use with any of these, especially if your slide rule sports the reminder ranges on the side of each.

Now even if you don't expect to need natural logs or exponentials, as covered in this and the previous installment, this has all been worth learning for what's to come next: computing arbitrary roots and powers. I think you'll be amazed at what's possible, once you understand the ranges of the LL scales.

Next installment: Logs of Arbitrary Base by LL Scales