First PrinciplesStart anywhere. Prove it, then move on.

Rung 9

Exponents and their inverse

Repeated multiplication, and the question that forces you to invent logarithms.

Best attempted after 5. Balancing equations. Nothing stops you trying this now — the gate will tell you if you were right.

The gate

Without a calculator, justify that 2^10 is a little over 1000, then use that one fact alone to estimate 2^20 and 2^30 and say how far off you expect to be. Then state roughly what power of 2 gives a million, and explain why that question is a logarithm.

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.

An exponent is repeated multiplication, which sounds too small to deserve a rung. It deserves one because of the question that follows: given the answer, what was the power? That question has no answer among the operations you already have, so a new one gets invented for it. That is what a logarithm is — not a button, an inverse.

Why this is on the ladder

Because growth by multiplication is the shape of almost everything that matters later: the number of states a system can be in, the number of paths through a network, how a quantity blows up or dies away, and — at rung 18 — the difference between an algorithm that finishes and one that does not.

2^10 ≈ 1000 is the single most useful number on this ladder. Memorise it, and you can estimate an enormous amount in your head.

Do this

Establish 2^10 = 1024 for yourself by doubling ten times. Write out every doubling; do not shortcut it.

Now use only that fact. 2^20 is 2^10 twice over, so a little over a million. 2^30, a little over a billion. Notice the error compounds: 1024 is 2.4% high, so 2^20 is about 5% high. Say how high before you check.

Then turn it around. Roughly what power of 2 reaches a million? You already know: about 20. You just computed a logarithm without the word.

Where people get stuck

2^3 · 2^4 gets multiplied into 2^12 rather than added into 2^7. The cure is not memorising the rule but writing both out as strings of 2s and counting. Anything you can rebuild from the definition, you cannot forget.

The other trouble is treating a logarithm as a new kind of object. It is a question, phrased as a number: what power? Every time you meet one, say the question out loud instead of reading the symbol.

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