the hook · may 22, 2010

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

principal P (BTC value, May 2010)$41.00annual rate r (BTC actual ≈ 189%)189%years t (since 2010)16

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

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 : . Click it. Drag r to 0.10 (SPY’s pace) and watch the future collapse — the same equation, a different story.

def future_value(P, r, t):
    return P * (1 + r) ** t

future_value(41, 1.89, 16)     # ≈ 1.0e9    (Laszlo today)
future_value(1e9, 1.89, 20)    # ≈ 1.7e18   (BTC in 2046, if it keeps pace)

How long? — when did the pizza hit $1M?

Reverse the question: at BTC’s actual rate, when did 10,000 BTC first cross ** $1,000,000**? You can't multiply your way there - you have to _undo_ the exponent. The <Term id="logarithm">logarithm</Term> is exactly that: given the result, _count the steps_.$ 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.

from math import log

def years_to_target(P, F, r):
    return log(F / P) / log(1 + r)

years_to_target(41, 1e6, 1.89)  # ≈ 9.5  (years since May 2010)

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.

def implied_rate(P, F, t):
    return (F / P) ** (1 / t) - 1

implied_rate(41, 1e9, 16)     # ≈ 1.89   (189% / yr — Bitcoin)
implied_rate(100, 1000, 30)   # ≈ 0.08   (8% — boring SPY-ish)

now break it

The single number “189% CAGR” hides 16 years of violence. Compute the CAGR over 2018 → 2022 (peak-to-peak): it’s around 3% per year. Compute it over 2020 → 2021: over 300%. CAGR is an average exponent — it assumes the rate was constant, which it never was. The same equation that discovered the Bitcoin story flattens it the moment you compute CAGR. If you sell at the wrong window, your “average” returns nothing like 189%.

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

P$41.00r189.0%t16 yr

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.)

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.)

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.

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.)

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?

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 $\log₁₀(2.89) ≈ 0.461$ .

your answer$

why this isn't taught this way

Most textbooks open with log(x) as “the inverse of eˣ” — a definition without a question. The reader learns the symbol before they learn the need. Lemma inverts the order: a ten-thousand-bitcoin pizza is the question, and exp/log/root fall out as the only honest way to answer it. The mathematics earns the page; the page does not earn the mathematics.

glossary · used on this page · 8

exponent·지수

The number of times you multiply a base by itself. In 2³ = 8, the exponent is 3.

⚠ 지수 also means 'index' (e.g. 주가지수 stock index). Same character, two roles.

logarithm·로그

The inverse of exponentiation. log_b(x) asks: what power of b gives x?

n-th root·거듭제곱근

The inverse of raising to the n-th power. The n-th root of x is x^(1/n).

⚠ '제곱근' alone means square root (n=2). For other n, say 'n거듭제곱근'.

Bitcoin (BTC)·비트코인

A digital currency launched in 2009 by an anonymous figure 'Satoshi Nakamoto'. New coins are released on a fixed schedule that halves every ~4 years.

compound interest·복리

Interest computed on principal plus accumulated interest. Each period multiplies, not adds.

CAGR·연평균 성장률

Compound Annual Growth Rate. CAGR = (F/P)^(1/t) − 1. The implied rate that turns P into F over t years.

SPY·SPY

The SPDR S&P 500 ETF — one of the largest exchange-traded funds. Tracks the S&P 500 (the 500 largest US public companies). Returns ~10% per year on average over the long run; used as the boring-but-honest benchmark in finance discussions.

⚠ Different from "the S&P 500" itself (which is just an index, not investable directly). SPY is one of several ETFs tracking the index — VOO and IVV are also common.

Rule of 72·72의 법칙

Doubling time ≈ 72 / (rate in %). At 8% per year, money doubles in ~9 years.

⚠ Why 72? ln(2) ≈ 0.693, and ln(1+r) ≈ r for small r. So t ≈ 0.693 / r. The 72 absorbs the small-r approximation error to land at a number with many divisors.