ACARA v9 CONTENT DESCRIPTION “interpret and use logarithmic scales in applied contexts involving small and large quantities and change”
Builds on: Exponential Relations (AC9M10A03). A logarithmic scale is the mirror of exponential growth: where exponentials multiply by a fixed factor each step, a log scale undoes that, turning equal multiplications into equal distances. The powers of ten met in exponential relations are exactly what these scales count.
A ruler that multiplies
Every scale met so far has been linear: the marks are equally spaced and equally valued, so one centimetre is the same length anywhere on a ruler, and the gap from 3 to 4 matches the gap from 30 to 31. A logarithmic scale works on a completely different principle. Its marks go 1, 10, 100, 1000, with each of these equally spaced, so that equal distances along the scale stand for equal multiplications rather than equal additions. Moving one step is not adding a fixed amount but multiplying by a fixed factor, usually ten. This sounds strange at first, but it is exactly the tool needed whenever the numbers you care about are spread across a huge range, from the very small to the very large.
Two rulers for the same numbers
A linear scale spaces numbers by equal amounts; a logarithmic scale spaces them by equal multiples.
On an ordinary linear scale, 1, 10 and 100 are squeezed into the far left while 1000 sits at the right, so the small values are unreadable. Switch to a logarithmic scale to see the fix.
Equal steps, equal multiplying
The single idea to hold onto is that on a logarithmic scale, a fixed distance always means multiplying by the same factor. If each step multiplies by ten, then one step takes you from 1 to 10, the next from 10 to 100, and the next from 100 to 1000, three equal steps covering a thousandfold change. Compare that with a linear scale, where reaching 1000 from 1 in three equal steps is impossible without enormous gaps. This multiplying behaviour is what gives a log scale its power to compress: a range that would need a sheet of paper kilometres long on a linear scale fits comfortably across a page, because each equal step swallows another factor of ten.
Equal steps mean equal multiplying
Moving a fixed distance on a logarithmic scale always multiplies the value by the same factor.
The defining feature of a logarithmic scale: equal distances stand for equal multiplications, not equal additions. Each step here is times ten, so moving three equal steps takes you from 1 all the way to 1000. This is why a log scale can show enormous ranges in a small space.
Why we need them: orders of magnitude
The reason logarithmic scales matter is that the world is full of quantities spanning many orders of magnitude, an order of magnitude being a factor of ten. The size of a bacterium and the size of a whale differ by around a hundred million times; the energy of a whisper and a jet engine, the populations of a village and a megacity, are similarly far apart. Plot such values on a linear scale and the small ones vanish into a single dot at zero beside the large ones, telling you nothing. A logarithmic scale rescues them by giving each factor of ten its own equal share of space, so the bacterium, the ant, the human and the whale all appear clearly on one diagram. Whenever data covers a vast range, the log scale is what makes it legible.
Fitting the tiny and the vast together
When values range over many orders of magnitude, a logarithmic scale shows them all at once.
A logarithmic scale shines when quantities span many orders of magnitude. The sizes of a bacterium, an ant, a human and a whale differ by a factor of a hundred million, yet a log scale places all four clearly. Switch to linear to see why a plain scale fails.
Logarithmic scales in the real world
Many familiar measurements are quietly logarithmic. The Richter scale for earthquakes adds one to its number for every tenfold increase in ground shaking, so a magnitude 7 is not a little worse than a 5 but a hundred times stronger. The decibel scale for sound, the pH scale for acids and bases, and the stellar magnitude scale for the brightness of stars all work the same way, each step standing for a fixed multiplying factor. Recognising that a scale is logarithmic is essential to reading it correctly, because the numbers must be interpreted as multipliers, not as ordinary amounts. A jump of two on such a scale signals a hundredfold change, a fact easy to miss and important to grasp.
The Richter scale is logarithmic
On the Richter scale each step of one means ten times the shaking, a logarithmic measure of earthquakes.
The Richter scale for earthquakes is logarithmic: each whole number up means ten times the ground shaking. So a magnitude 5 shakes ten times as hard as a 4, and a magnitude 7 a hundred times as hard as a 5. That is why a difference of just two points marks the gap between a tremor and a disaster, and why these scales need to be read multiplicatively.
Reading a logarithmic scale
Because a log scale multiplies, it must be read with care, especially between the marked values. On a linear scale the point halfway between 1 and 10 is 5.5, the ordinary average, but on a logarithmic scale halfway in distance means halfway in multiplying, which lands at about 3.16, the square root of ten. Treating a log scale as if it were linear is the classic mistake and gives badly wrong readings. The habit to build is to think multiplicatively: ask how many times larger, not how much larger. Once that shift is made, logarithmic scales become a natural and powerful way to picture quantities that would otherwise be impossible to show together, which is precisely why scientists and engineers reach for them so often.
Reading between the marks
Halfway along a logarithmic scale is the geometric middle, found by multiplying, not the ordinary average.
A point sits halfway between the 1 and 10 marks. On a linear scale that would be 5.5, but a log scale works by multiplying. Reveal its true value.
Quick self-check
1. On a logarithmic scale, equal distances represent:
2. Logarithmic scales are especially useful for quantities that:
3. On the Richter scale, a magnitude 7 earthquake shakes the ground how much harder than a magnitude 5?
4. On a logarithmic scale marked 1, 10, 100, the point halfway between 1 and 10 represents about: