Maths That Predicts Behaviour
Logarithms & Decibels
12 min · 2 worked examplesStudent plan
Why gain is measured in dB, how a logarithm compresses a huge range into a readable one, and the numbers worth memorising.
After this lesson you can
- Explain what a logarithm actually does to a number
- Convert a power ratio into decibels and back
- Read a log axis without misjudging it by an order of magnitude
A logarithm asks "what power?"
log₁₀(1000) = 3 asks: ten to what power gives 1000? The answer is 3. That is all a logarithm is, the inverse of raising to a power.
The useful consequence is that logarithms turn multiplication into addition. log(a × b) = log(a) + log(b). Multiplying by ten adds one to the log; multiplying by a million adds six.
This is why log scales appear wherever a quantity spans an enormous range. Human hearing covers a factor of about a trillion in sound power. Plotted linearly, everything quiet is crushed into a flat line at the bottom. Plotted logarithmically, quiet and loud both get readable space.
In the full lesson
- A logarithm asks "what power?"
- The decibel
- Where you will meet this
This lesson comes with the Student plan. The first module of Maths for Engineers is free.