TwiddleLab / Math / Growth: Exponents and Logs

Faster and Faster: the Number e

Lesson 10 of 11 · 7 min · Free, no account needed

What you will learn

• Compound interest pays interest on interest: 50% twice turns $1 into $2.25, not $2.
• Splitting 100% into n payments gives (1 + 1/n)ⁿ, which approaches e ≈ 2.71828.
• Growth paid continuously is e^(rt). Its inverse is the natural log ln: ln 2 ≈ 0.69, so 6.9% doubles in 10 years.

Start the lesson

How the lesson goes

1. A bank pays 100% a year. Would you rather get it once, or 1% a hundred times?

   Once turns $1 into $2. Bet what 1% a hundred times gives.

2. Start small: two payments of 50%.

   Pay the interest twice and watch the stack.

3. Split the 100% into more and more payments.

   Turn the dial from once a year to every second.

4. The ceiling is a number: e.

   (1 + 1/n)ⁿ grows with n but never passes e ≈ 2.71828.

   Growth paid continuously at rate r for time t multiplies by e^(rt).

   The inverse of eˣ is the natural log, ln x: ln e = 1, ln 2 ≈ 0.693.

5. Make $1 double in exactly 10 years.

   The bank pays continuously, so $1 becomes e^(rt). Turn the rate until it lands on $2.

   Doubling means e^(10r) = 2. The natural log undoes e: 10r = ln 2 ≈ 0.693, so r ≈ 6.9%. That is where the rule of 70 comes from: 70 / rate ≈ years to double.

6. Game: beat the bank.

   Four rounds. Tap the bank that turns $1,000 into more in one year.

7. AI uses e in the sigmoid.

   The sigmoid 1 / (1 + e^(−x)) squeezes any number into 0 to 1. Drag the point until the output is 0.9.

   1/(1 + e^(−x)) = 0.9 when e^(−x) = 1/9, so x = ln 9 ≈ 2.2.

8. $1,000 at 100% a year, paid continuously. How much after one year?

   Paying 100% continuously multiplies by e^(1 × 1) = e ≈ 2.71828, so $1,000 becomes about $2,718.28.

9. One more thing that makes e special.

   Tap a curve to see its tangent at x = 0. Which one has slope exactly 1? Then Check.

   The slope of bˣ at 0 is ln b: ln 2 ≈ 0.69, ln 3 ≈ 1.10, and ln e = 1. That is why e is the natural base.

Previous lesson: Log Scales · Next lesson: Project: Will It Go Viral? · Part of the course: Growth: Exponents and Logs

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