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

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**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 log _{e} (ln) or log_{10}:**

- Linearising the exponential growth/decay equation,
*N*=_{t}*N*_{0}e^{(kt)}

**Applications in science:**

- Radioactivity decay - Half life (see Use of growth/decay factors for
*T*_{1/2}= -0.693 /*k*)

- Bacterial decay- Decimal reduction time (see Use of growth/decay factors for
*T*_{D }= -2.303 /*k*)

- Bacterial growth - Generation time (see Use of growth/decay factors for
*T*_{G}= +0.693 /*k*)

- Capacitor discharge - Time constant (see Use of growth/decay factors for
*τ*= -1.0 /*k*)

QVA Tutorials (questions + video answers):

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.

- Choosing the correct method of analysis (
**Example quiz**)

Background Theory

Background Theory

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

*N _{t}* =

*N*

_{0}e

^{(kt)}which can also be written for clarity as

*N*=

_{t}*N*

_{0}exp(

*kt*)

This equation gives the value of the variable, *N*_{t}, at at time, *t*, as a function of time.

*N*_{0} is the value of *N*_{t} when *t* = 0, and *k* is the growth (or decay) constant which determines how quickly the value of *N*_{t} is increasing (or decreasing).

If *k* is positive then *N*_{t} is increasing, but if *k* is negative, then *N*_{t} is decreasing.

The process of linearisation involves taking the **natural logarithms (log _{e} or ‘ln’) of both sides** of the equation:

ln{*N*_{t}} = ln{*N*_{0}exp(*kt*)}

The RHS can be modified using the fact that the logarithm of the product - *N*_{0} times exp(*kt*) - will be the addition of the separate lograrithms

ln{*N*_{t}} = ln{*N*_{0}} + 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{*N*_{t}} = ln{*N*_{0}} + *kt*

It is now useful to rearrange the RHS slightly:

ln{*N*_{t}} = *k* × *t* + ln{*N*_{0}}

which now looks like a straight line equation:

*y *= *m *× *x* + *c*

where

ln{*N*_{t}} 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{

*N*

_{0}} would give the intercept,

*c*, of the line.

Plotting ln{*N _{t}*} 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 *N*_{0} calculated from ** N_{0} = e^{c}**.

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

log{*N*_{t}} = 0.4343 × *k* × *t* + log{*N*_{0}}

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 *N*_{0} **when using logs to base 10 **are given by:

*k* = *m* / 0.4343 = 2.303 × *m* and *N*_{0} = 10* ^{c}*.