Notes on Using Logarithms

Important equations involving logarithms

This section gives the key equations involving logarithms and exponentials that are used in the different methods of analysis for non-linear data.

• Natural logarithm is the inverse of the power of 'e':
If ep = q  then  p = loge(q)= ln(q)

• Logarithm to base 10 is the inverse of the power of '10':
If 10p = q  then  p = log10(q) = log(q)

• Logarithm of a product:
ln(p×q) = ln(p) + ln(qand  log(p×q) = log(p) + log(q)

• Taking the logarithm of its inverse:
ln(ep) = p  and  log(10p) = p

• Taking the logarithm of a general exponential:
ln(pq) = q × ln(pand log(pq) = q × log(p)

Taking logarithms of data variables

There is a technical issue with taking the logarithms of experimental data values.
This does not cause any real problems, except that it is essential that the units of measurement being used are the same on both sides of the equation.

It is only possible to take the logarithm of a pure number, e.g. log(5.62) = 0.750
It is not possible to take the logarithm of a data variable which has units, e.g. distance, d = 5.62 m, has units of metres.

However, when we take logarithms of both sides of an equation, the units will be the same on both sides of the equation and can be considered to cancel, leaving only numeric values.
For example, in the equation for distance, d, speed, v and time t:
d = v × t
the velocity, v, must be expressed in the same units as d and t, e.g.
5.62 m = 2.81 m s-1 × 2.0 s
The units of metres and seconds on either side ‘cancel’ leaving the pure numeric values:
5.62 = 2.81 × 2.0
We can now take logarithms of the numeric values on both sides of this equation:
log(5.62) = log(2.81 × 2.0)

When we say that we are taking the logarithm of the distance we are effectively taking the logarithm of the numeric value of the distance. We must then be very careful that the units we have used for measurement are consistent on both sides of the equation.