First PrinciplesStart anywhere. Prove it, then move on.

Rung 10

Logarithmic scale

Why a straight line on a log axis means something is multiplying.

Best attempted after 9. Exponents and their inverse. Nothing stops you trying this now — the gate will tell you if you were right.

The gate

Take real data spanning at least three orders of magnitude, plot it on a linear axis and on a log axis, and write the one sentence each plot supports and the other cannot. Then state what a straight line on the log axis proves about the underlying process — and what it does not.

Nobody checks this but you. Do it honestly and the rungs above hold; do it loosely and they will not, somewhere further up where the cause is much harder to find.

Change the axis and the same numbers tell a different story. On a linear axis, equal steps mean equal addition: 10, 20, 30. On a log axis, equal steps mean equal multiplication: 10, 100, 1000. Nothing about the data changed. What changed is which pattern is visible.

Why this is on the ladder

Because you will spend rungs 44 to 48 staring at a loss curve, and it will be on a log axis, and reading it wrongly will cost you hours. A loss that looks like it has flattened on a linear plot is often still falling steadily — by a constant factor — which is exactly what a log axis reveals and a linear one hides.

More generally: a straight line on a log axis is the visual signature of constant multiplication. Once you can see it, you see it everywhere.

Do this

Get real data that spans orders of magnitude. Populations of towns and cities. File sizes on your machine (you can produce that list with rung 8's script). Anything where the biggest value is a thousand times the smallest.

Plot it both ways. On the linear plot, almost everything will be squashed against zero and only the largest values will be legible — that is not a flaw, it is the honest consequence of the scale, and it is the sentence that plot supports.

Now the careful part. A straight line on the log axis says the quantity is multiplying by a constant factor per step. It does not say why, and it does not licence extrapolating forward. Say both.

Where people get stuck

People read the log plot as if the axis were still linear, and conclude that growth slowed down when it did not. Read the gridlines before the curve.

The subtler error is treating a straight log-line as proof of an exponential process. Plenty of things look straight over a short range. The plot is evidence, not a mechanism — and the difference between those two is a habit worth having long before rung 38 makes it formal.

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