TwiddleLab / Math / Growth: Exponents and Logs
Log Scales
What you will learn
- On a log scale, each equal step multiplies by the same number, usually 10.
- Earthquake magnitude, decibels and pH are log scales: +1 magnitude is ×10 wave, +10 dB is ×10 sound power.
- Exponential growth is a straight line on a log axis; its slope is the growth factor.
How the lesson goes
An earthquake of magnitude 7 vs magnitude 5. Is it 'a bit' stronger?
Bet how many times taller the wave on the seismograph is.
Turn up the magnitude.
Each step on the dial makes the wave 10 times taller. The paper zooms out to fit it.
Straighten a growth curve.
A colony doubles every hour. Drag the axis lever to log, then drag the two blue handles so the line runs through the dots.
A log axis spaces 1000, 10,000, 100,000 equally. Doubling every hour multiplies by the same factor each hour, so every hour climbs the same distance: a straight line.
A log scale counts multiplications.
On a log scale the position is the log of the number: equal steps mean equal multiplications.
Richter magnitude: +1 is ×10 wave height. Decibels: +10 dB is ×10 sound power. pH: one less is ×10 more acidic.
Exponential growth looks straight on a log axis. Its slope tells the growth factor.
Put the sounds on the decibel ladder.
A whisper is 30 dB. Each sound card says how many times more powerful it is than a whisper.
Game: log darts.
Five numbers. Drag the dart to where each one sits on the log axis, then Throw.
AI labs plan big models on log-log plots.
On log axes, error against compute is a straight line. Turn the dial to 10,000 times the compute.
Use your line: when does the colony reach 1,000,000?
Drag the flag to the hour, then Check.
The colony doubles every hour from 1000. A million is 1000 times more, and 2¹⁰ ≈ 1000, so it takes about 10 hours. On a log axis that is where the straight line meets the million line.
The earthquake energy ladder.
Tap each magnitude to see its energy compared with magnitude 4.
Previous lesson: Log Rules: Exponent Rules Backwards · Next lesson: Faster and Faster: the Number e · Part of the course: Growth: Exponents and Logs
Lessons are interactive and need JavaScript. Turn it on to play them.
