3A Logarithms = logs / 3B Quick logs

Chapter 3 · pages 78–83 · 1300 words

This section names the single most useful re-expression tool: the logarithm, shortened throughout to log. Building on the earlier observation (from the power-plant stem-and-leaf in chapter 1) that batches of always-positive numbers whose values are not too close together tend to have their leaves crowded into the low digits, the book argues that taking logs is the first re-expression to try, with square roots as the second choice. The key property that makes logs useful is that equal differences in logs correspond to equal ratios of the raw values — so taking logs turns products into sums and ratios into differences, which is why logs help whenever people work with ratios or products (price indexes, growth rates, and so on). Whenever a kind of log is needed other than base 10 (common logarithms), the text promises to flag it specially; otherwise "log" always means base-10.

The worked example re-does the Bureau of Reclamation hydroelectric-plant data. In raw megawatts the leaves were washed into the corner where leading digits are 1 or 2; taking logs spreads the batch out and makes it far more symmetric. A compressed version of that log stem-and-leaf, unit 0.1, looks like this (leaves are the second digit after the decimal point of the log):

depth | stem | leaf
  1   | 1*   | 1                (smallest plant)
  5   |  t   | 2333
 11   |  f   | 445555
 16   |  s   | 66677
 (2)  |  .   | 88
 16   | 2*   | 0000011
  9   |  t   | 23
  7   |  f   | 445
  4   |  s   | 6                San Luis
  3   |  .   | 9                Glenn Canyon
  2   | 3*   | 1                Hoover
  1   |  t   | 3                Grand Coulee

Both the extremes, hinges, and median are reported in both logs and the original megawatts side by side, because when working with a re-expressed batch it is good practice to show letter values in both scales — this keeps the "feel" for the raw units while doing the analysis in the more manageable, more symmetric log scale.

Section 3B turns to the practical question of how to get logs fast without a full-precision table or a calculator's worth of decimals. Two-decimal logs — and often even one-decimal logs — are precise enough for almost any re-expression job; the guiding rule is to use two-decimal logs first, and only fall back to many-decimal logs in the rare case where more precision is truly needed. To make two-decimal logs fast, the book introduces the break table (what others might call a critical table): a table that tells you which of a small number of possible log values a raw number's leading digits falls between, so you just find where your number lies between two "breaks" and read off the log's fractional part directly, with no interpolation. When a value lands exactly on a break, the convention is to round to an even answer.

An example applies the method: for the Corps of Engineers hydroelectric data, a value beginning "14" falls between the breaks 1396 and 1429, so its two-decimal log ends in .15; a value beginning "18" falls between 1799 and 1841, giving a log ending in .26. Applying this systematically to the whole batch turns a raw stem-and-leaf whose leaves pile up at low leading digits into a log stem-and-leaf that is longer and more "gangling" but gives, as the text puts it, a far more honest picture of how the values are spread out — confirming that the raw display, useful mainly for storage, gave little help for visualization compared to the log version.

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