Wren pins last week's lap-time sheet to the wall. "The crew's model: lap time is (3s + 20) over (s + 2), for s sessions."
"I want a 5 minute lap," Comet says. "Which session gets me there? What can we make of this rule backwards?"
"Same moves," Wren says. "Swap, then solve for y. The fraction just takes more steps."
He works it on the board. "Session 5. Check: 3 times 5 plus 20, over 5 plus 2, is 5."
Nova dims the dock lights one click at a time. "Would you like a hint? My glow meter reads 3, then 6, then 12."
"Doubling," Comet says. "Reading equals 3 times 2 to the clicks. Backwards, that asks which exponent."
"Which exponent is a logarithm," Wren says. "The inverse of an exponential rule. Let us write both."
Put in a lap time and get back the session. L⁻¹(5) = 5, so session 5 gives Comet her 5 minute lap.
The inverse has a wall of its own at x = 3. No session gives a 3 minute lap, which matches week 3's asymptote.
| Sessions (s) | Lap time L(s), minutes | Backwards |
|---|---|---|
| 0 | 10 | L⁻¹(10) = 0 |
| 2 | 6.5 | L⁻¹(6.5) = 2 |
| 5 | 5 | L⁻¹(5) = 5 |
| 8 | 4.4 | L⁻¹(4.4) = 8 |
| 18 | 3.7 | L⁻¹(3.7) = 18 |
The dimmer rule is g(x) = 3 × 2ˣ: start at 3, then double once per click. Its last step is "multiply by 3."
To undo it, divide by 3 first, then ask which exponent of 2 gives that number. That question is log₂.
So g⁻¹(x) = log₂(x/3). For a reading of 48: 48 ÷ 3 = 16, and log₂ 16 = 4. 4 clicks.
A logarithm is an exponent. The sentence log_b N = k says the same as b to the k = N, roles swapped.
| Clicks (x) | Glow reading g(x) | Backwards |
|---|---|---|
| 0 | 3 | log₂(3/3) = 0 |
| 1 | 6 | log₂(6/3) = 1 |
| 2 | 12 | log₂(12/3) = 2 |
| 3 | 24 | log₂(24/3) = 3 |
| 4 | 48 | log₂(48/3) = 4 |
| 5 | 96 | log₂(96/3) = 5 |
Both ways work for every rule. Swap-and-solve is safest when the rule is a fraction. Undo-the-steps is quickest when each step is plain.
| Statement | True or false? |
|---|---|
| log₂ 16 = 4 because 2⁴ = 16. | ? |
| The inverse of g(x) = 3 × 2ˣ is log₂(3x). | ? |
| 3 × 2 × 2 × 2 × 2 = 48 | ? |
| A logarithm is an exponent. | ? |
| L⁻¹(x) = (-2x + 20)/(x - 3) has no value at x = 3. | ? |
Clear work. Tomorrow is Boathouse Lab: a cardboard tube becomes a winch, and your own table runs backwards.