Earthquakes, loudness, star brightness — all read on a log scale.
A stock chart that rose 100x over 30 years — why do the first 20 years look like they're crawling along the floor?
A lens that compresses an exploding world to human scale — the log function.
Experiment
Hands-on experiment
Predict first — a stock rose at the same rate (~17%) every year for 30 years. Which was the better investment: the first 10 years or the last 10?
A 30-year stock (~17%/yr)
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Plot an exponentially growing value on a linear scale and the early years get crushed flat against the floor. A stock jumping 100→200 (a doubling!) is the same '2x' as 10,000→20,000, yet it looks completely different.
To see fairly, you need a scale where 'how many times bigger' shows as equal spacing — the log scale, where 10, 100, and 1000 sit at equal intervals.
On a log scale, exponential growth becomes a straight line. You stop being fooled by the bend of the curve and read the growth rate itself — this is why log functions are essential literacy in the data age.
Insight
Insights from the video
“A straight line on a log scale means growth at a constant ratio.”
The slope of a log chart is the growth rate. A straight line means the same % growth every year; a bend in the slope means the growth rate changed. It's a powerful skill for reading graphs in the news.
“Exponential and logarithm are two names for the same relationship, seen from opposite sides.”
2¹⁰ = 1024 and log₂1024 = 10 are the same sentence. Reflect the graph of y=aˣ in the mirror y=x and the log function appears — not a new function, but a reversal of viewpoint.
Misconception
Common misconceptions
A log chart is a distortion that shrinks changes.
It's the linear scale that distorts, by flattening the large early growth rates. A doubling always means the same thing, yet the linear scale draws it differently each time. The log scale is what shows the 'world of ratios' fairly.
Log functions grow slowly, so they're not that important.
Growing slowly is precisely its superpower. Pass any explosive value through a log and it becomes a manageable size — which is why information content, entropy, pH, decibels, and star magnitudes are all defined with logs.
Formula
Writing it as math
Let's write what you saw in the scale-toggle experiment as formulas.
Definition of the log function
The exponential function with its input and output swapped. That's why the two graphs are symmetric about y=x.
Why exponential growth becomes a line
Take the log of both sides and you get a linear expression in x — a straight line. The slope (log a) is the growth rate.
How slowly logs grow
Even with an input of a million, the output is just 6. This extreme compression is why logs became the standard tool in every field that handles big numbers.
In Real Life
Where you meet it in real life
Long-run stock charts
Long-term investment analysis is always done on log charts. Same slope = same return, so gains in the 1980s and the 2020s can be compared fairly.
Reading epidemic graphs
On a log scale, a straight case-count curve means exponential spread; a bending slope means the spread is slowing — the graphs in pandemic news were logs.
The scale of our senses
Sound (dB), star brightness, earthquake magnitude — all log units, because human perception itself responds to the log of the stimulus (the Weber–Fechner law).
Algorithm efficiency
Binary search finds one item among a million in about 20 steps because log₂(1,000,000) ≈ 20. Computer science talks about performance in logs.
Connection
Concepts connect
Previous concept
Exponential Functions
Only after living through the exploding curve do you appreciate the lens that tames it.
← Exponential Functions labLeads to next
Logarithms
The root of the log function — head to the Log Lab, which handles the question 'how many multiplications?' itself, and watch multiplication turn into addition.
Go to the Logarithms lab →