College Algebra / Unit 8 Alg 473 worked example, 60 to 90 seconds
Doubling Time
The idea
Doubling time is the t solving 2 = e^(rt) under continuous growth — a fixed span for a fixed rate, independent of the starting amount.
In practice
A population grows at 6% continuously; find its doubling time. Set the ratio to 2: 2 = e^(0.06t). Take the natural log of both sides — ln is e's own inverse, so it peels the exponential in one move: ln(2) = 0.06t. Divide: t = ln(2)/0.06 ≈ 0.6931/0.06 ≈ 11.55 years. The audit runs forward: e^(0.06 · 11.55) ≈ e^0.693 ≈ 2. And the independence claim earns its keep: no starting population appeared anywhere — the P canceled when the ratio was set to 2, which is why one rate owns one doubling time for towns of any size.
Where people slip
Match the log to the base — ln for e — and the peel takes one move instead of three.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
Builds on
- Nothing. This is a starting point.
Unlocks
- Nothing yet depends on this.
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