Logarithms and log scales

An introduction

Most of the material selection charts are plotted using logarithmic scales. We’re not going to go into the details of the maths of logarithms here, but will just give an idea of how and why we use them.

Logarithms are related to ‘times’ scales - for example in the Richter scale for earthquakes, an increase in 1 point on the scale corresponds to an increase of 10 times more energy released. The decibel scale used for sound is another common example of a log scale.

So we need a scale which looks something like this:

We can see that there is an easy relationship between the linear scale and the log scale - if the point on the linear scale is n then the equivalent point on the log scale is 10n. So point ‘X’ on the scale above is about 100.5 = 3.16

The reverse of this procedure is going from the log scale to the linear scale. If the point on the log scale is p, then the equivalent point on the linear scale is log(p). You can find the log function on most calculators (note it is related to, but not the same as, the ln function). So 2 on the log scale is at point log(2)=0.30 on the linear scale, 3 on the log scale is at log(3)=0.48 on the linear scale etc. Use a calculator to check that you can go from a linear scale to a log scale and back again to the same point on the linear scale. In maths terms this means that log(10n)=n

Looking at the log scale between 1 and 10, we can calculate where all the intervals lie and plot them using the linear scale:

And we get a similar pattern between 10 and 100 (without showing the linear scale this time!):

On all the log graphs we use, we mark these intervals for you to make them easier to read without a calculator:

You can see from this scale that the higher values are ‘squashed’ towards the right-hand end and the lower values ‘spread out’ towards the left-hand end. This behaviour is useful when we are looking at material properties. Let’s take the range of Young’s modulus for 2 materials and say it varies by a factor of 2. For material A the range is between 2GPa & 4GPa and for material B the range is between 200GPa & 400GPa. Plotting these on a linear axis shows us:

Since the values for B are so much greater, the values for A can barely be seen on this linear scale, which is not much use! However, plotting the same ranges on a log scale reveals what we’re looking for much more clearly:

As most material properties cover large ranges, it is sensible to plot them using log axes so we can see the property ranges for individual materials more clearly.

Note that on the scale above, the distance between the upper and lower limits for A is the same as the distance for B. Both correspond to a "factor of 2". This is always the case on a log scale, two points on the scale at a given spacing have the same factor between them, wherever you put the two points. It is particularly easy to identify factors of 10, 100, 1000 and so on since the main divisions on the log scale indicate these "factors of 10".

All the log scales on the charts show factors of 10, with one exception: resistivity. This covers so many factors of 10 (24 from top to bottom of the chart) that the axis would be very cluttered if they were all marked. The chart therefore shows factors of 1,000,000 instead. Remember that there are really 5 intermediate factors of 10, spaced equally between division on the scale. Factors of 10-100 therefore look small, but are important in selecting materials for electrical conductors.