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Logarithmic scales
Intermediate
Log axes turn power laws into straight lines.
A log axis spaces values by powers of ten. Power laws y = axⁿ become straight lines on log–log axes with slope n, and exponentials become straight lines on semi-log axes. Decibels and the Richter scale are logarithmic.
A 20 dB increase in sound level means a 100-fold increase in intensity.
Equal distances on a log axis are equal ratios, not equal differences, and zero cannot appear on it.
In more detail
A logarithmic scale spaces values by their exponent, so each step represents a multiplication by ten rather than an addition. Logs are used to compare quantities spanning many orders of magnitude, such as sound intensities or earthquake energies.
Real-world uses
- Decibels measure sound level on a logarithmic scale because hearing responds to ratios of intensity.
- The Richter and moment magnitude scales compare earthquake sizes across many orders of magnitude.
- Earthquake magnitudes and sound levels in decibels use logarithmic scales, so a range of many orders of magnitude fits on one chart.
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Frequently asked questions
What is a base-ten logarithm?
log10 of x is the power you raise ten to in order to get x, so log10 of 1000 is 3.
Why is a 10 dB increase a big deal?
It means the sound intensity is ten times greater, so the logarithmic scale compresses a huge range into a manageable number.
Why is an increase of 10 dB ten times more power?
Decibels are logarithmic, so each 10 dB step multiplies the power by ten. A 20 dB increase is a hundredfold increase in power.
Related topics in Measurement & Math Tools
Complex numbers and phasors · Graphs and slopes · Curve fitting and linearisation