## 3.5 The Log Scale – L

Many slide rules have the L scale on them, which gives the logarithm of the number aligned typically on the C or D scale. As can be seen from our plot labeled `Logarithmic Scale`

in Logarithms and Log Scales, if the values on the C scale have a logarithmic spacing, then the logarithms of those values will have a linear spacing. Hence, the L scale on a slide rule is easy to identify as it has its digits 1 to 10 spaced evenly by about one inch on a standard 10 inch rule.

An early use of the L scale was to find the value of a number raised to an arbitrary power. Suppose you wanted to know \(3.7^{4.6}\). Then one could find the log of 3.7 = 0.568 on the L scale and multiply by 4.6 to get 2.614 using the C and D scales. Then find on the L scale which number has the logarithm 0.614 – which is 4.11 – and then multiply by 100 (due to the “2” out front in 2.614) to get the final answer: 411. Check by computer:

`## [1] 3.7 to the power 4.6 = 410.89223180408 .`

We noted in previous sections that sometimes reading squares and cubes from the A and K scales can yield results only accurate to 1-2 digits. If a more accurate answer is required, the L scale often can be used to perform the calculation. For example, let’s re-compute the volume of a sphere of radius \(R\) = 4.58 inches. Rather than using the K scale, which yielded “about 400” cubic inches, we do the following:

- From our formula for the volume of a sphere, \(V = \frac{4\pi}{3}R^3\), we see that \(\log V = \log (4\pi/3) + 3 \log R\).

- If a gauge mark is not already on the rule, find our necessary constant of \(4\pi/3\) using the C/D scales. It should be about 4.19.

- Using the L scale, find the log of 4.58, which is about 0.661, and the log of 4.19, which is about 0.622.

- Using C/D, or mentally, find 3 times 0.661, which should be roughly 1.983.

- Adding these results, 0.622 + 1.983 = 2.605 = 0.605 + 2.

- Resetting the slide if necessary, use the L scale to determine the number whose log is 0.605. I find about 4.03; the “2” in the above result tells us to use 403. A more precise answer, using a personal computer, is 402.425.

We see that using the L scale for this purpose may give more accuracy, but it also involves many more steps and perhaps the writing down of intermediate values. If one only needs a more approximate answer, the use of the A/B and K scales can save significant time.