For instance, 42 = 16 can be rewritten as log4 16 = 2, although you would say “log to the base 2 of 16 is 4.” Log is the exponent, in this example, 4. This allows the separation of the exponent on one side of an equation. “Logs” are another way of writing exponential equations. For example, 10 and 20 compared to 80 and 90 are not equally spaced apart instead 10 and 100 and 60 and 600 will be equally spaced apart because they both represent a 100 percent increase. Factors of 10-100 therefore look small, but are important in selecting materials for electrical conductors.Logarithms are non-linear. Remember that there are really 5 intermediate factors of 10, spaced equally between division on the scale. The chart therefore shows factors of 1,000,000 instead. 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. 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".Īll the log scales on the charts show factors of 10, with one exception: resistivity. 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. Note that on the scale above, the distance between the upper and lower limits for A is the same as the distance for B. 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:Īs 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. Plotting these on a linear axis shows us: For material A the range is between 2GPa & 4GPa and for material B the range is between 200GPa & 400GPa. Let’s take the range of Young’s modulus for 2 materials and say it varies by a factor of 2. This behaviour is useful when we are looking at material properties. 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. On all the log graphs we use, we mark these intervals for you to make them easier to read without a calculator: Looking at the log scale between 1 and 10, we can calculate where all the intervals lie and plot them using the linear scale:Īnd we get a similar pattern between 10 and 100 (without showing the linear scale this time!): In maths terms this means that log(10 n)=n 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. 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. You can find the log function on most calculators (note it is related to, but not the same as, the ln function). If the point on the log scale is p, then the equivalent point on the linear scale is log(p). The reverse of this procedure is going from the log scale to the linear scale. So point ‘X’ on the scale above is about 10 0.5 = 3.16 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 10 n.
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