TwiddleLab / Math / Growth: Exponents and Logs
Log Rules: Exponent Rules Backwards
What you will learn
- log₂ x is the inverse of 2ˣ: its graph is 2ˣ reflected in y = x.
- log(xy) = log x + log y and log(x/y) = log x − log y: exponent rules, read backwards.
- log(xᵏ) = k · log x. That is why a slide rule, and a computer adding log probabilities, can multiply by adding.
How the lesson goes
Before calculators, engineers multiplied huge numbers by adding. How?
Start with two ordinary rulers. Slide the top one to add 2 + 3.
Now the marks are doublings: 1, 2, 4, 8...
Slide the top ruler so its 1 sits over 4. Read what is under the top's 8.
Why does this work? Flip 2ˣ over.
Tap each point of 2ˣ. It flips across the line y = x.
Exponent rules, backwards.
log₂ x is the inverse of 2ˣ: 2³ = 8, so log₂ 8 = 3. Its graph is 2ˣ flipped in y = x.
2ᵃ · 2ᵇ = 2ᵃ⁺ᵇ turns into log(xy) = log x + log y, and dividing into log(x/y) = log x − log y.
(2ᵃ)ᵏ = 2ᵃᵏ turns into log(xᵏ) = k · log x. These work for log₁₀ too.
A real slide rule: 2 × 3.5.
Here the marks are spaced by log₁₀, so 1 to 10 fills the ruler. Slide the top's 1 over 2 and read under 3.5.
Game: slide rule sprint.
Four products. Slide the top's 1 over the first number; the answer appears under the orange line.
How likely is a 150-word sentence?
A language model gives each word a probability, here 0.001. Machine A multiplies them. Machine B adds their logs. Feed both the sentence.
Multiplying many small probabilities gives a number smaller than a computer can store, so it becomes 0. The log of the product is the sum of the logs, −3 per word, which stays an ordinary number.
Grade Pip's log homework.
Tap every wrong line, then Check.
Log rules come from exponent rules: multiplying adds exponents, dividing subtracts them, powers multiply them. There is no exponent rule for adding, so log(2 + 8) does not split.
Make 32 from log 2 lengths.
Lay lengths of log 2 end to end on the doubling ruler until you land on 32.
Previous lesson: Logs Count Doublings · Next lesson: Log Scales · Part of the course: Growth: Exponents and Logs
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