Maths for A Level Chemistry - updated edition

We have the following general rule: If X = b p then by deϐinition, p = log b X . It is important to note the following: log 10 1 = 0 because 10 0 = 1 and log 10 10 = 1 because 10 1 = 10 Note also that you cannot ϐind the log of zero or the log of a negative number. Bases used for logarithms There are two bases used for logarithms, base 10 and base e where e is the number 2.71828... In A-level Chemistry we only need to deal with logarithms having a base 10. Log 10 X is written simply as Log X, so the base is left out and it is assumed it is 10. To ϐind the log of a number using a calculator, simply press the button marked log and then enter the number and press =. Using logs in relation to quantities that range over several orders of magnitude You can see that logs are a bit complicated, so why are they used? Logarithmic scales are scales of measurement used where the value of a physical quantity uses intervals that correspond to orders of magnitude rather than a standard linear scale. Suppose we wanted to draw a graph and instead of plotting a linear scale such as 0, 0.1, 0.2, 0.3, 0.4, 0.5 on one of the axes we needed to plot the following values 1, 10, 100, 1000, 10,000. If you try to draw this on a set of axes you will see the problem. As the scale varies in orders of magnitude (i.e. multiples of 10 in this case), it will be difϐicult as some of the values such as 0, 1, 10 and 100 are very close together when the scale has to reach 10,000. To get around this problem we can use a logarithmic scale where we ϐind the log of each of the numbers. We would now have a scale which covers log 1, log 10, log 100, log 1000 and log 10 000. Now, ϐinding the logs of the numbers would give a scale covering 0, 1, 2, 3 and 4. We have now reduced the scale to a much more manageable size. The presentation of data using a log scale is very useful when the data covers a large range of values. Later on in the book you will come across the pH scale, which is used to represent the acidity, neutrality or alkalinity of a solution. The quantity used to calculate pH (i.e. the hydrogen ion concentration) varies over many orders of magnitude, so this is why a logarithmic scale is used. 3 Algebra 37