Logarithmic Scales
Theory
On a logarithmic scale, equal steps stand for multiplying by the same number, usually \(10\), so very small and very large quantities fit on one axis. This Year 10 maths page shows how to read log scales on axes, compare quantities using orders of magnitude, and use the Richter, pH and decibel scales and doubling scales such as musical octaves.
On a linear scale, equal steps add the same amount: \(0, 10, 20, 30, \ldots\)
On a logarithmic scale (log scale), equal steps multiply by the same number. On a base-\(10\) log scale the marks are \(1, 10, 100, 1000, \ldots\), that is \(10^0, 10^1, 10^2, 10^3, \ldots\), equally spaced. Going one step right multiplies by \(10\); one step left divides by \(10\). Numbers less than \(1\) sit to the left of \(1\): \(0.1 = 10^{-1}\), \(0.01 = 10^{-2}\). There is no zero on a log scale.
The order of magnitude of a number is the power of \(10\) it is closest to from below. A number between \(10^n\) and \(10^{n+1}\) has order of magnitude \(n\), so \(4.7 \times 10^5\) has order of magnitude \(5\). Two quantities that differ by \(2\) orders of magnitude differ by a factor of roughly \(100\).
Between two major marks, the minor marks stand for \(2, 3, 4, \ldots, 9\) times the major mark on their left. They are not evenly spaced: they bunch up towards the next power of \(10\).
Log scales are used when quantities cover several powers of \(10\): earthquake magnitude (Richter scale), acidity (pH) and sound level (decibels, dB). A log scale is also used on graphs of money or populations that grow by the same percentage each period.
Two readings on a base-\(10\) log scale that are \(d\) steps apart stand for quantities in the ratio
\[\text{ratio} = 10^{d}\]| Scale | A change of | means |
|---|---|---|
| Richter magnitude | \(+1\) | \(10\) times the ground movement |
| pH | \(-1\) | \(10\) times as acidic |
| decibels | \(+10\) dB | \(10\) times the sound intensity |
| musical note | \(+1\) octave | \(2\) times the frequency |
How to compare two readings
- Subtract the readings to find how far apart they are.
- Change to steps. For magnitude and pH one unit is one step; for decibels divide by \(10\).
- Multiply: the ratio is \(10^{\text{steps}}\).
- Say which is bigger. Higher magnitude or more dB is bigger; for pH, the lower reading is more acidic.
Working backwards
- Write the factor as a power of \(10\): \(1000 = 10^3\) is \(3\) steps, \(\dfrac{1}{100} = 10^{-2}\) is \(2\) steps the other way.
- Add or subtract those steps (\(10\) dB per step for sound) from the reading you know.
Reading a log axis
- Find the major gridlines (powers of \(10\)) either side of the point.
- Count the minor marks: the \(k\)th minor mark after \(100\) is \((k + 1) \times 100\).
- Equal distances mean equal factors. Moving the same distance anywhere on the axis multiplies by the same number, so a quantity that grows by the same percentage every year plots as a straight line.
Choosing a scale
- Use a log scale when the values span several orders of magnitude; on a linear scale the small values would be squashed together near zero.
- Use a linear scale when the values are of similar size or when you need to read differences directly.
(i) How many times the ground movement of a magnitude \(4.1\) earthquake is that of a magnitude \(7.1\) earthquake?
(ii) Another earthquake has one-thousandth of the ground movement of a magnitude \(7.3\) earthquake. What is its magnitude?
(i) Find the number of steps, then multiply by \(10\) for each one.
| \(d\) | \(=\) | \(7.1 - 4.1 = 3\) |
| \(\text{ratio}\) | \(=\) | \(10^3 = \mathbf{1000}\) |
(ii) Write the factor as a power of \(10\). Dividing by \(1000 = 10^3\) is \(3\) steps down the scale.
| \(\text{magnitude}\) | \(=\) | \(7.3 - 3\) |
| \(\) | \(=\) | \(\mathbf{4.3}\) |
Check: from \(4.3\) to \(7.3\) is \(3\) steps, and \(10^3 = 1000\). ✓
(ii) A garden party is \(75\) dB and a jet taking off nearby is \(135\) dB. How many times as intense is the jet's sound?
(iii) A sound is \(1000\) times as intense as a \(50\) dB sound. What is its sound level?
(i) The lower pH is more acidic: stomach acid. Each step down the scale is \(10\) times as acidic.
| \(9 - 2\) | \(=\) | \(7 \text{ steps}\) |
| \(\text{ratio}\) | \(=\) | \(10^7 = \mathbf{10\,000\,000}\) |
(ii) Divide the difference by \(10\) to get the steps.
| \(135 - 75\) | \(=\) | \(60 \text{ dB} = 6 \text{ steps}\) |
| \(\text{ratio}\) | \(=\) | \(10^6 = \mathbf{1\,000\,000}\) |
(iii) Work backwards: \(1000 = 10^3\) is \(3\) steps of \(10\) dB.
| \(L\) | \(=\) | \(50 + 3 \times 10 = \mathbf{80}~\textbf{dB}\) |
(ii) Between which two powers of \(10\) do \(0.003\) and \(48\,000\) lie?
(i) Compare the powers of \(10\), then divide for the exact factor.
| \(30 - 24\) | \(=\) | \(\mathbf{6} \text{ orders of magnitude}\) |
| \(\dfrac{2 \times 10^{30}}{6 \times 10^{24}}\) | \(=\) | \(0.333\ldots \times 10^{6}\) |
| \(\) | \(\approx\) | \(\mathbf{3.3 \times 10^{5}}\) |
The Sun is about \(330\,000\) times as heavy, which is close to \(10^6\) divided by \(3\).
(ii) Write each number in scientific notation and read the power.
| \(0.003\) | \(=\) | \(3 \times 10^{-3}\) |
| \(\) | \(\) | \(\text{between } \mathbf{10^{-3}} \text{ and } \mathbf{10^{-2}}\) |
| \(48\,000\) | \(=\) | \(4.8 \times 10^{4}\) |
| \(\) | \(\) | \(\text{between } \mathbf{10^{4}} \text{ and } \mathbf{10^{5}}\) |
(ii) How many octaves apart are notes of \(30\) Hz and \(1920\) Hz?
(iii) The number of bacteria in a culture is graphed against time on a log scale. The graph is a straight line from \(50\) bacteria at hour \(0\) to \(50\,000\) bacteria at hour \(6\). How many bacteria were there at hour \(4\)?
(i) Each octave multiplies by \(2\).
| \(f\) | \(=\) | \(65 \times 2^4\) |
| \(\) | \(=\) | \(65 \times 16 = \mathbf{1040}~\textbf{Hz}\) |
(ii) Divide to find the ratio, then write it as a power of \(2\).
| \(\dfrac{1920}{30}\) | \(=\) | \(64\) |
| \(64\) | \(=\) | \(2^6\) |
So the notes are \(\mathbf{6}\) octaves apart.
(iii) On a log axis a straight line means the number is multiplied by the same factor in equal times.
| \(\dfrac{50\,000}{50}\) | \(=\) | \(1000 = 10^3\) |
That is \(3\) factors of \(10\) in \(6\) hours, so the number is multiplied by \(10\) every \(2\) hours. Hour \(4\) is \(2\) steps after hour \(0\).
| \(N\) | \(=\) | \(50 \times 10^2\) |
| \(\) | \(=\) | \(\mathbf{5000}\) |
Adding instead (\(50\,000 \div 6\) per hour) would treat the axis as linear, which it is not.
Common pitfalls
Frequently asked questions
What is a logarithmic scale?
A scale where equal steps multiply by the same number instead of adding the same amount. On a base-\(10\) scale the marks are \(1, 10, 100, 1000\) and so on, equally spaced.
Why use a logarithmic scale?
When values cover several powers of \(10\), a linear scale squashes the small values together near zero. A log scale gives each power of \(10\) the same space, so small and large values can be read on one axis.
How many times stronger is a magnitude 8 earthquake than a magnitude 5?
The magnitudes differ by \(3\), so the ground movement is \(10^3 = 1000\) times as large.
What does order of magnitude mean?
The power of \(10\) a number sits at or above. \(4.7 \times 10^5\) has order of magnitude \(5\). Quantities that differ by one order of magnitude differ by a factor of roughly \(10\).
What is a doubling scale?
A scale where each equal step multiplies by \(2\) instead of \(10\). Musical octaves work like this: each octave up doubles the frequency, so \(4\) octaves up is \(2^4 = 16\) times the frequency.