Logarithmic Scales (decibels, seismic, star magnitude, pH)
Logarithmic scales use \(\log\) to compress very large ranges into workable numbers — decibels for sound, the Richter scale for earthquakes, star magnitude and pH for acidity.
Part of the NSW Year 12 Mathematics Advanced course, in the Functions area of study (Further graph transformations and modelling focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
Logarithmic scales compress quantities that range over many orders of magnitude into manageable numbers, where each step of 1 means a tenfold change. This Year 12 Mathematics Advanced topic (MAV-12-02) covers pH, decibels and the Richter scale, using base-10 logarithms.
A logarithmic scale records \(\log_{10}\) of a raw quantity (up to constants), so a difference of \(k\) units on the scale means a ratio of \(10^{k}\) in the quantity. This suits data spanning many orders of magnitude.
Common scales: \(\text{pH}=-\log_{10}[\mathrm{H}^{+}]\); the Richter magnitude from wave amplitude; and decibels \(L=10\log_{10}\!\dfrac{I}{I_0}\).
Method
- Pick the right formula (pH, decibels, Richter).
- Substitute the raw quantity and evaluate the base-10 log (or reverse it with a power of 10).
- Compare using: a difference of \(k\) scale units \(=\) a factor of \(10^{k}\).
| \(\text{pH}\) | \(=\) | \(-\log_{10}(10^{-3})\) |
| \(=\) | \(3\) |
| \(\text{difference}\) | \(=\) | \(6-4=2\) |
| \(\text{ratio}\) | \(=\) | \(10^{2}=100\) |
| \(\log_{10}[\mathrm{H}^{+}]\) | \(=\) | \(-5\) |
| \([\mathrm{H}^{+}]\) | \(=\) | \(10^{-5}\text{ mol/L}\) |
| \(L\) | \(=\) | \(10\log_{10}(10^{6})=60\text{ dB}\) |
| \(20\) | \(=\) | \(10\log_{10} r\Rightarrow r=100\) |
Common pitfalls
Frequently asked questions
What is a logarithmic scale?
A logarithmic scale measures the log of a quantity, so each step of one unit means the quantity is ten times bigger. It is used when values range over many orders of magnitude, such as sound intensity or earthquake size.
How do you calculate pH from hydrogen ion concentration?
Use pH equals minus log base 10 of the hydrogen ion concentration. For example if the concentration is 10 to the minus 3, the pH is 3.
How much bigger is a magnitude 6 earthquake than a magnitude 4?
A difference of 2 on the Richter scale means the wave amplitude is 10 to the power 2, which is 100 times larger.
Why are logarithmic scales useful?
They compress a huge range of values into a small, readable set of numbers, so quantities differing by factors of thousands or millions can be compared easily.