A Florida programmer named Laszlo Hanyecz paid 10,000 BTC for two Papa John's pizzas — about $41.
Sixteen years later, those 10,000 BTC are worth about $1 billion. The most expensive meal in history. So: what annual rate turns $41 into $1B in 16 years? When did Laszlo's dinner first cost a million dollars? And — the one that bites — how much would it be worth in 2046 if BTC keeps that pace?
You cannot answer any of these without three operations: exp, log, and n-th root. They are not three topics. They are three doors of one room — and the same machine that discovers a story like Bitcoin's also detects when a story is impossible. This page is the room.
Widget A — The Pizza Slider
Laszlo's $41 (P)$41.00
annual rate r189.0%
years t16 yr
F = P(1+r)t$970.84M
year reached2026
Try this: drop r to 10% (SPY's long-run rate). The line on the log axis goes shallow — barely tilted. Crank r back to 189%. It steepens. The slope on a log axis is the rate. You're reading exponential growth as one number: how steep.
the arc
1
How much? — projecting the pizza forward
If Bitcoin keeps its historical pace, what's 10,000 BTC worth in 2046? Stack ×2.89 twenty more times on $1B. Compound growth is repeated multiplication; repeated multiplication, compressed into one symbol, is exponentiation. F = P · (1 + r)t. Tap "Code" mode at the top — three lines of Python.
2
How long? — when did the pizza hit $1M?
Reverse the question: at BTC's actual rate, when did 10,000 BTC first cross $1,000,000? You can't multiply your way there — you have to undo the exponent. The logarithm is exactly that: given the result, count the steps. t = log(F / P) ÷ log(1 + r). Log is the inverse of exp. They are defined by each other. Answer: about 9.5 years in — late 2019. The pizza became a million-dollar pizza on a random Tuesday.
3
What rate? — was Laszlo's loss really 'extraordinary'?
We know $41 became $1B in 16 years. We don't know r. We can't undo the exponent (we don't know it); we can't take a log (we'd get the wrong unknown). We need a third operation: the n-th root. r = (F / P)1/t − 1. Finance calls this CAGR. It is a fractional exponent — a 1/16th power. The number that comes out: 189% per year. SPY does ~10%. Buffett's Berkshire (his investment company): ~20%. NVIDIA over 25 years: ~33%. Bitcoin: 189. Roots are exponents whose value is not a whole number.
One equation, three operations.F = P(1+r)t has three unknowns. Each unknown picks a different door: exp for F, log for t, root for r. That is the entire structural relationship between exp, log, and root. Memorize the equation, not the operations.
Widget B — Three Doors
F = P · (1 + r)t
operation: exponent
F (future value)
$970.84M
One equation. Three unknowns. exp isolates F. log isolates t. root isolates r. Same machine, three doors.
exercises · 손으로 풀기
1read the graph
On the Pizza Slider above, leave it at the defaults: P = $41, r = 189%. Toggle the y-axis to log. Roughly when does Laszlo's $41 first cross $1,000,000? (Read it off the graph; no formula.)
About 2019 — roughly 9–10 years in. The straight log-line meets the $1M grid line near year 9.5. (Sanity check: $41 × 24,000 = $984k, and 2.899.5 ≈ 24,000, so 9.5 years checks.)
2compute by hand · Rule of 72no calculator
The Rule of 72 at SPY's 8% per year: how long for $1 to double? Then justify the rule using ln(2) ≈ 0.693 and ln(1 + r) ≈ r for small r. Then ask yourself: would the rule still work at BTC's 189%? (Don't compute — just predict.)
72 / 8 = 9 years. Justification: (1+r)t = 2 ⇒ t = ln(2) / ln(1+r) ≈ 0.693 / r. So t·(r in %) ≈ 69.3, rounded up to 72 (lots of divisors). At BTC's 189%, the approximation ln(1+r) ≈ r breaks badly — the "72" rule predicts doubling in 72/189 ≈ 0.38 years (≈4.5 months), but the true answer is ln(2)/ln(2.89) ≈ 0.65 years (≈8 months). The rule is a small-r tool. Pin this lesson — exercise 6 will revisit it.
3write the equation
You buy $100 of BTC today. Assume — generously — that BTC's CAGR slows to 50% per year (about a quarter of its historical pace). Write the equation for its value in 10 years, then solve.
F = 100 · 1.510. Compute 1.510: 1.5² = 2.25, 1.5⁴ = 5.06, 1.5⁸ = 25.6, 1.510 = 1.5⁸ · 1.5² = 25.6 · 2.25 ≈ 57.7. So F ≈ $5,770. The "boring" half-pace future still puts $100 at almost $6k in a decade. That's what compounding does even when slowed.
4compute by hand · the rootno calculator
A friend put $1,000 into BTC 5 years ago. It is now $32,000. What was their CAGR? (Hint: notice 32 = 2⁵.)
r = (32000/1000)1/5 − 1 = 321/5 − 1 = (2⁵)1/5 − 1 = 2 − 1 = 100% per year. Their portfolio doubled every year. The trick was spotting that 32 is 2⁵, so the 5th root collapses to a clean 2.
5two curves cross
Investor A puts $1,000,000 into SPY at 8%/yr in 2010. Investor B puts just $100 into BTC at 50%/yr in 2010 (sober assumption, not the 189% historical). After how many years does B catch up to A?
Set 100 · 1.5t = 1,000,000 · 1.08t. Divide: (1.5/1.08)t = 10,000, so 1.389t = 10,000. Take log10: t · log10(1.389) = 4. With log10(1.389) ≈ 0.143, t ≈ 4 / 0.143 ≈ 28 years — i.e. around 2038. Lesson: in compounding, rate beats principal, but only if you give it enough time. The break-even is set by the log of the principal gap.
6the evil one · log as lie detector
Project Bitcoin forward. Today (2026): 10,000 BTC ≈ $1B. If BTC keeps its 16-year CAGR of 189% for another 20 years (until 2046), what is the pizza worth? Compare to global GDP (~$110T). What does the answer tell you about exponential stories? Use log10(2.89) ≈ 0.461.
F = 10⁹ · 2.89²⁰. Take log10: log10(F) = 9 + 20 · 0.461 = 9 + 9.22 = 18.22. So F ≈ 1018.22 ≈ $1.7 × 1018 — about $1.7 quintillion. Global GDP is ~$1.1 × 1014. Ratio: ~15,000× global GDP. The lesson is the punchline of this whole module: exponentials cannot survive contact with finite resources. The same log that discovered Bitcoin's 189% rate also predicts when that rate becomes nonsense. Logs aren't just inverse exponents. They are the bullshit detector for any story told in compounding.