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 √3e3τ-87 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 "t" then "h" then "e" then "o" then "r" then "e" then "m" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Jul 2020

Happy √3e3τ-87 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

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

Archive

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