A Chart With a Squashed Axis
Last timeTwo, Ten, and the Awkward One
On a logarithmic axis equal distances are equal factors, not equal amounts. Reading one correctly turns growth into a slope, and misreading one is how charts mislead honestly.
A quantity that grows by multiplying cannot be drawn on an ordinary chart. Not
with difficulty: at all. If the smallest value is one and the largest is a
million, then any scale that fits the million into the picture puts the one
indistinguishably on the baseline, along with the thousand and the ten
thousand, and the first nine tenths of the story is a flat line on the floor.
The fix is to plot the logarithm instead of the quantity, and then label the
axis with the original values so the reader never has to do the conversion.
That is all a logarithmic axis is.
Why anyone bothers
Taking the log of the quantity fixes it, because the log of something that
multiplies by a fixed factor each step gains a fixed amount each step. That is
the product rule from two lessons ago, used on a chart.
| step 0 | step 4 | step 8 | step 12 | step 16 | step 20 | |
|---|---|---|---|---|---|---|
| the amount | 1.00 | 16.00 | 256.00 | 4096.00 | 65536.00 | 1048576.00 |
| its base ten log | 0.00 | 1.20 | 2.41 | 3.61 | 4.82 | 6.02 |
| gain in the log since th | 0.00 | 1.20 | 1.20 | 1.20 | 1.20 | 1.20 |
Two straight lines worth recognising
Almost all of the value in reading these charts comes from two patterns, and
both are a straight line. Which pattern it is depends on how many of the axes
are squashed.
A straight line with the vertical axis logarithmic and the horizontal axis
ordinary means multiplicative growth: the quantity is multiplying by a fixed
factor per unit of the horizontal axis. The slope is the rate. The useful
summary to take off the chart is how far along you must travel to climb one
decade, because that is the time to grow tenfold, and dividing it by about
3.32 gives the doubling time.
A straight line with both axes logarithmic means a power relationship: one
quantity is a fixed power of the other. The slope is the exponent, read
directly, with no conversion. A slope of two means the vertical quantity goes
as the square of the horizontal one; a slope of minus a half means it falls as
one over the square root.
The lesson stops here
2 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents