Learn about log vs linear price scales — Stock-Market notes

WHAT is a price scale?

WHY does this matter? (Derivation from first principles)

Let price be $P$ and screen-height be $y$ .

Linear rule. We choose $y$ to be proportional to price: $y = kP$ Then a move from $P_1$ to $P_2$ has screen distance $\Delta y = k(P_2 - P_1).$ So equal $\$ moves → equal distance. A $10 move looks the same everywhere. *Why is this bad for long charts?* Because a$ 10 move is a huge deal at $20 (+50\%) but trivial at$ 2000 (+0.5%), yet they look identical.

Log rule. Now choose $y$ to be proportional to the logarithm of price: $y = k\ln P.$ The screen distance of a move is $\Delta y = k(\ln P_2 - \ln P_1) = k\ln\!\frac{P_2}{P_1}.$

Consequence for exponential growth. Suppose a stock compounds at rate $r$ per year: $P(t) = P_0\,e^{rt}.$ On a linear axis this curves upward faster and faster (unreadable early history). On a log axis: $y = k\ln P(t) = k\ln\!\big(P_0 e^{rt}\big) = k\ln P_0 + k r\,t.$

Figure — Learn about log vs linear price scales

HOW to read them in practice

Common mistakes (Steel-man → Fix)

Recall Feynman: explain to a 12-year-old

Imagine a ladder. On a normal (linear) ladder, every rung is 1 foot apart — good for measuring how many feet you climbed. But money doesn't grow by feet, it grows by multiplying. So we build a magic (log) ladder where each rung means "twice as much money" instead of "one more dollar." On this magic ladder, going from $1 to$ 2 is the same size step as going from $1000 to$ 2000 — because both doubled your money. That's why grown-ups use the magic ladder for charts that span many years: it shows how much richer you got in percent, not just in dollars.

Active-recall flashcards

#flashcards/stock-market

What does equal distance represent on a linear price scale?

Equal absolute (dollar/point) change.

What does equal distance represent on a log price scale?

Equal percentage (ratio) change.

On a log scale, screen distance of a move equals what expression?

k\ln(P_2/P_1)

— depends only on the price ratio.

Why does constant % growth appear as a straight line on a log chart?

Because

y=k\ln(P_0 e^{rt}) = k\ln P_0 + kr\,t

, which is linear in

t

with slope

kr

On a log chart, what does the slope of a straight trend line tell you?

The growth rate

r

(times constant

k

20\to40

200\to240

: which is the bigger move on a log scale?

20\to40

(+100%,

\ln2=0.693

) beats

200\to240

(+20%,

\ln1.2=0.182

Why do log and linear look similar over tiny ranges?

Because

\ln(1+x)\approx x

for small

x

, so % moves ≈ $ moves.

For decades-long index history, which scale should you use and why?

Log, so early and recent eras are comparably visible as percent moves rather than being dwarfed by large recent numbers.

On a log chart the gridlines 10, 100, 1000 are spaced how?

Equally — each step is ×10.

Does a log chart change the actual price data?

No — only the axis spacing; the plotted prices are real prices.

Connections

Charts, Trends & Dow Theory
Support and Resistance — trendlines are more reliable on the correct scale
Dow Theory — trends over long horizons are best read on log scale
Compound Annual Growth Rate (CAGR) — $P=P_0 e^{rt}$ is the log-linear model
Percentage vs Absolute Returns
Reading Long-Term Index Charts

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, chart ke y-axis (price wale side) ko do tareeke se draw kar sakte ho. Linear scale me equal dollar/point ka move barabar height leta hai — yaani $10 se$ 20 aur $100 se$ 110 dono same lambai dikhte hain. Log scale me equal percentage move barabar height leta hai — $10 se$ 20 (double) aur $1000 se$ 2000 (double) dono ek jaisa dikhte hain. Kyunki hum paisa percent return me sochte hain (10% profit chhota ho ya bada stock, feel same hota hai), isliye long-term charts ke liye log scale hi sahi aur honest scale hai.

Maths simple hai: log scale par ek move ki height $\ln(P_2/P_1)$ hoti hai — sirf ratio pe depend karti hai, absolute price cancel ho jaati hai. Aur agar stock constant rate $r$ se compound ho raha hai ( $P = P_0 e^{rt}$ ), to log chart par woh ek seedhi line ban jaata hai, jiska slope growth rate batata hai. Isliye log chart par straight line ka matlab hai "steady percent growth" — steepen hone ka matlab return badh raha hai.

Practical baat: short-term ya chhote range me log aur linear almost same lagte hain (kyunki $\ln(1+x)\approx x$ ). Lekin jab decades ka data ho ya price 10x-100x badh gaya ho, tab linear chart purana history dabaa deta hai aur recent bade numbers hi bade dikhte hain — yeh galat impression deta hai. Isliye investors long-term aur index charts hamesha log scale par dekhte hain, taaki har era ka percentage move fairly compare ho sake.

Yaad rakho: LINE = dollars, LOG = ratio. Trendlines, support-resistance, sab tab reliable hote hain jab scale sahi choose kiya ho. Compounding assets ke liye woh log scale hai.

Test yourself — Charts, Trends & Dow Theory

Connections