## Using Log-Lin Plots for Exponential Data - This page is in the process of development

### Video Tutorials

These short step-by-step video tutorials show the detailed process of linearising exponential data, together with real applications from different branches of science.

Core techniques using loge (ln) or log10:

Applications in science:

### QVA Tutorials (questions + video answers):

These short quizzes (5 - 10 questions each, with video feedback) provide a quick way of checking and increasing your knowledge and real understanding of the linearising of exponential data.

### Background Theory

A simple growth or decay relationship is usually expressed in the form of:

Nt = N0e(kt) which can also be written for clarity as Nt = N0exp(kt)

This equation gives the value of the variable, Nt, at at time, t, as a function of time.
N0 is the value of Nt when t = 0, and k is the growth (or decay) constant which determines how quickly the value of Nt is increasing (or decreasing).
If k is positive then Nt is increasing, but if k is negative, then Nt is decreasing.

The process of linearisation involves taking the natural logarithms (loge or ‘ln’) of both sides of the equation:

ln{Nt} = ln{N0exp(kt)}

The RHS can be modified using the fact that the logarithm of the product - N0 times exp(kt) - will be the addition of the separate lograrithms

ln{Nt} = ln{N0} + ln{exp(kt)}

The RHS can be modified using the fact that the process of ln and exp are the inverse of one another so 'ln of exp(kt)' just goes back to the starting value, kt.

ln{Nt} = ln{N0} + kt

It is now useful to rearrange the RHS slightly:

ln{Nt} = k × t + ln{N0}

which now looks like a straight line equation:

y = m × x + c

where
ln{Nt} is equivalent to the variable y,
t is equivalent to the variable x,
the constant, k, would give the slope (gradient), m, of the line, and
the value of ln{N0} would give the intercept, c, of the line.

Plotting ln{Nt} against t gives a straight line, and Excel can be used to calculate the slope, m, and intercept, c, of this straight line.

The growth constant can then be calculated directly, k = m,
and N0 calculated from  N0 = ec.

It is also possible to use logs to base 10 instead of natural logs, giving

log{Nt} = 0.4343 × k × t  +  log{N0}

The only difference is that the slope in the equation is now equal to 0.4343 times k,
where 0.4343 = log(e) to 4 sig fig.

The calculations for k and N0 when using logs to base 10 are given by:
k = m / 0.4343 = 2.303 × m  and N0 = 10c.