## 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 e^{p}=*q***then***p*= log_{e}(*q*)= ln(*q*)

**Logarithm to base 10 is the inverse of the power of '10'**:

If 10^{p}=*q***then***p*= log_{10}(*q*) = log(*q*)

**Logarithm of a product**:

ln(*p×q*) = ln(*p*) + ln(*q*)**and**log(*p×q*) = log(*p*) + log(*q*)

**Taking the logarithm of its inverse**:

ln(e^{p}) =*p***and**log(10^{p}) =*p*

**Taking the logarithm of a general exponential**:

ln(*p*^{q}) =*q*× ln(*p*)**and**log(*p*^{q}) =*q*× log(*p*)

Taking logarithms of data variables

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.