mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

 2020-03-31 
Recently, you've probably seen a lot of graphs that look like this:
The graph above shows something that is growing exponentially: its equation is \(y=kr^x\), for some constants \(k\) and \(r\). The value of the constant \(r\) is very important, as it tells you how quickly the value is going to grow. Using a graph of some data, it is difficult to get an anywhere-near-accurate approximation of \(r\).
The following plot shows three different exponentials. It's very difficult to say anything about them except that they grow very quickly above around \(x=15\).
\(y=2^x\), \(y=40\times 1.5^x\), and \(y=0.002\times3^x\)
It would be nice if we could plot these in a way that their important properties—such as the value of the ratio \(r\)—were more clearly evident from the graph. To do this, we start by taking the log of both sides of the equation:
$$\log y=\log(kr^x)$$
Using the laws of logs, this simplifies to:
$$\log y=\log k+x\log r$$
This is now the equation of a straight line, \(\hat{y}=m\hat{x}+c\), with \(\hat{y}=\log y\), \(\hat{x}=x\), \(m=\log r\) and \(c=\log k\). So if we plot \(x\) against \(\log y\), we should get a straight line with gradient \(\log r\). If we plot the same three exponentials as above using a log-scaled \(y\)-axis, we get:
\(y=2^x\), \(y=40\times 1.5^x\), and \(y=0.002\times3^x\) with a log-scaled \(y\)-axis
From this picture alone, it is very clear that the blue exponential has the largest value of \(r\), and we could quickly work out a decent approximation of this value by calculating 10 (or the base of the log used if using a different log) to the power of the gradient.

Log-log plots

Exponential growth isn't the only situation where scaling the axes is beneficial. In my research in finite and boundary element methods, it is common that the error of the solution \(e\) is given in terms of a grid parameter \(h\) by a polynomial of the form \(e=ah^k\), for some constants \(a\) and \(k\).
We are often interested in the value of the power \(k\). If we plot \(e\) against \(h\), it's once again difficult to judge the value of \(k\) from the graph alone. The following graph shows three polynomials.
\(y=x^2\), \(y=x^{1.5}\), and \(y=0.5x^3\)
Once again is is difficult to judge any of the important properties of these. To improve this, we once again begin by taking the log of each side of the equation:
$$\log e=\log (ah^k)$$
Applying the laws of logs this time gives:
$$\log e=\log a+k\log h$$
This is now the equation of a straight line, \(\hat{y}=m\hat{x}+c\), with \(\hat{y}=\log e\), \(\hat{x}=\log h\), \(m=k\) and \(c=\log a\). So if we plot \(\log x\) against \(\log y\), we should get a straight line with gradient \(k\).
Doing this for the same three curves as above gives the following plot.
\(y=x^2\), \(y=x^{1.5}\), and \(y=0.5x^3\) with log-scaled \(x\)- and \(y\)-axes
It is easy to see that the blue line has the highest value of \(k\) (as it has the highest gradient, and we could get a decent approximation of this value by finding the line's gradient.

As well as making it easier to get good approximations of important parameters, making curves into straight lines in this way also makes it easier to plot the trend of real data. Drawing accurate exponentials and polynomials is hard at the best of times; and real data will not exactly follow the curve, so drawing an exponential or quadratic of best fit will be an even harder task. By scaling the axes first though, this task simplifies to drawing a straight line through the data; this is much easier.
So next time you're struggling with an awkward curve, why not try turning it into a straight line first.

Similar posts

Visualising MENACE's learning
World Cup stickers 2018, pt. 2
Happy 3e√3π-43 Approximation Day!
A surprising fact about quadrilaterals

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "i" then "n" then "t" then "e" then "g" then "e" then "r" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Jul 2020

Happy 3e√3π-43 Approximation Day!

May 2020

A surprising fact about quadrilaterals
Interesting tautologies

Mar 2020

Log-scaled axes

Feb 2020

PhD thesis, chapter ∞
PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

raspberry pi pi asteroids sound graphs estimation royal institution hannah fry signorini conditions graph theory oeis preconditioning pythagoras football sport sorting harriss spiral interpolation golden spiral braiding arithmetic noughts and crosses chalkdust magazine advent calendar royal baby final fantasy rhombicuboctahedron weather station gerry anderson twitter tennis error bars gaussian elimination palindromes phd go london logs christmas card chebyshev dates php speed video games draughts folding paper simultaneous equations chess game show probability data martin gardner pizza cutting bubble bobble approximation machine learning christmas ternary propositional calculus world cup dataset nine men's morris captain scarlet manchester science festival a gamut of games binary news programming computational complexity hats light reddit dragon curves inline code matrix of cofactors flexagons weak imposition rugby london underground cambridge triangles inverse matrices python map projections convergence national lottery coins accuracy electromagnetic field manchester matt parker boundary element methods probability frobel matrix of minors geogebra books data visualisation realhats games golden ratio exponential growth latex tmip pac-man european cup reuleaux polygons numerical analysis polynomials game of life matrix multiplication folding tube maps statistics bempp fractals plastic ratio wool people maths puzzles quadrilaterals the aperiodical craft determinants mathsteroids pi approximation day trigonometry menace stickers javascript platonic solids sobolev spaces geometry squares talking maths in public mathsjam ucl hexapawn radio 4 countdown finite element method matrices cross stitch bodmas big internet math-off mathslogicbot logic misleading statistics wave scattering curvature

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020