This subsection relates to the description of log-linear plots covered in Section 6.6.
Frequently in engineering, the question arises of how the parameters of an exponential function might be found from given data. The method follows from the fact that it is possible to ‘undo’ the exponential function and obtain a linear function by means of the logarithmic function. Before showing the implications of this method, it may be necessary to remind you of some rules for manipulating logarithms and exponentials. These are summarised in Table 1 on the next page, which exactly matches the general list provided in Key Point 8 in Section 6.3 (page 22.) Table 1: Rules for manipulating base logarithms and exponentials
We will try ‘undoing’ the exponential in the particular example
We take the natural logarithm ( ) of both sides, which means logarithm to the base . So
The result of using Rule 1a in Table 1 is
The natural logarithmic functions ‘undoes’ the exponential function, so by Rule 4a,
and the original equation for becomes
Compare this with the general form of a linear function .
If we regard as equivalent to , 0.1 as equivalent to the constant as equivalent to , and as equivalent to the constant , then we can identify a linear relationship between and . A plot of against should result in a straight line, of slope 0.1, which crosses the axis at . (Such a plot is called a log-linear or log-lin plot.) This is not particularly interesting here because we know the values 12 and 0.1 already.
Suppose, though, we want to try using the general form of the exponential function
to create a continuous model for a population for which we have some discrete data. The first thing to do is to take logarithms of both sides
Rule 1 from Table 1 then gives
But, by Rule 4a, , so this means that
So, given some ‘population versus time’ data, for which you believe can be modelled by some version of the exponential function, plot the natural logarithm of population against time. If the exponential function is appropriate, the resulting data points should lie on or near a straight line. The slope of the straight line will give an estimate for and the intercept with the axis will give an estimate for You will have carried out a logarithmic transformation of the original data for . We say the original variation has been linearised .
A similar procedure will work also if any exponential function rather than the base exponential function is used. For example, suppose that we try to use the function
where A and B are constant parameters to be derived from the given data. We can take natural logarithms again to give
Rule 1a from Table 1 then gives
Rule 3a then gives
Again we have a straight line graph with the same intercept as before, , but this time with slope .
The amount of money to which grows after earning interest of 5 p.a. for years is worked out as
Find a linearised form of this equation.
Take natural logarithms of both sides.
Rule 3b gives
So a plot of against would be a straight line passing through with slope
The linearisation procedure also works if logarithms other than natural logarithms are used. We start again with
and will take logarithms to base 10 instead of natural logarithms. Table 2 presents the laws of logarithms and indices (based on Key Point 8 page 22) interpreted for .
Table 2 : Rules for manipulating base 10 logarithms and exponentials
Taking logs of gives:
Rule 1a from Table 2 then gives
Use of Rule 3a gives the result
Write down the straight line function corresponding to taking logarithms of the
general exponential function
by taking logarithms to base 10.
- Write down the slope of this line.
It is not usually necessary to declare the subscript 10 when indicating logarithms to base 10. If you meet the term ‘log’ it will probably imply “to the base 10”. In the remainder of this Section, the subscript 10 is dropped where is implied.