Semi-log Plots (College Board AP® Precalculus): Study Guide

Semi-log plots

What is a semi-log plot?

• A semi-log plot is a graph in which one of the axes uses a logarithmic scale

  • instead of a standard (linear) scale

• On a logarithmically scaled axis, each unit of distance represents a multiplicative change rather than an additive one

  • E.g. on a standard y-axis, the marks might be at 0, 50, 100, 150, 200, ...

  • On a log-scaled y-axis (base 10), the marks are at 1, 10, 100, 1000, ...

    • each step up represents multiplication by 10

• The logarithmically scaled axis can use any base $n$, where $n>1$

  • Base 10 is the most common choice

  • The natural base $e$ is also sometimes used

• In this course, the semi-log plots you encounter will have the y-axis logarithmically scaled and the x-axis on a standard scale

Why do exponential functions appear linear on a semi-log plot?

• An exponential function $y=a{b}^x$ has outputs that grow by a constant multiplicative factor ($b$) for each unit increase in $x$

  • On a logarithmically scaled y-axis, equal multiplicative changes correspond to equal vertical distances

  • So an exponential function, which involves repeated multiplication, produces equally spaced points vertically

    • i.e. the data appears linear on the semi-log plot

• One key rule to remember:

  • If data points appear to lie along a straight line on a semi-log plot (with logarithmic y-axis)

  • then the data can be modeled by an exponential function

• E.g. consider the data in the following table

• Note that the ratio of successive terms (45/25, 81/45, etc.) is approximately 1.8 in each case

  • This should already let you know that an exponential model would be appropriate!

• On a standard plot, these points form a steeply curving exponential curve

• On a semi-log plot (log-scaled y-axis), the same points appear to lie along a straight line

• Note the uneven spacing of the horizontal grid lines on the semi-log graph

  • The lines between 1 and 10 (on the vertical axis) correspond to $y=2,3,4,...,9$

  • The lines between 10 and 100 correspond to $y=20,30,40,...,90$

  • The lines between 100 and 1000 correspond to $y=200,300,400,...,900$

• This sort of unevenly spaced grid is a key sign that a graph is a semi-log plot

What is the advantage of a semi-log plot over checking ratios?

• You can test for exponential behavior by checking whether the ratios of successive output values are constant

  • However, this ratio test requires an additive constant to be removed first if the data has a vertical shift

    • e.g. if $y=a{b}^x+k$

• A semi-log plot has the advantage of allowing exponential behavior to be spotted

  • without having to adjust dependent variable values by a constant first

• Specifically, if

  • for large input values of a data set

  • the logarithms of the output values trend linear

    • i.e. if the output values lie (approximately) along a straight line on a semi-log plot

  • then transformations of an exponential function can be used to model the data

• This works because for a shifted exponential like $y=a{b}^x+k$

  • the exponential term $a{b}^x$ dominates for large $x$

  • so the additive constant $k$ becomes negligible

• The logarithms of the output values will therefore trend toward a linear pattern as $x$ increases

  • even without removing $k$ first

• This makes semi-log plots a more robust visual test for exponential behavior than the ratio method

Linearising exponential data

How does the linearization work mathematically?

• For an exponential model $y=a{b}^x$

  • taking the logarithm (base $n$) of both sides

    • and using properties of logarithms

  • gives

$\begin{array}{rcl}{\log }_{n}y& =& {\log }_n\left(a{b}^x\right)\\ & =& {\log }_{n}a+{\log }_n\left(b^x\right)\\ & =& {\log }_{n}a+x{\log }_{n}b\end{array}$

• This has the form $\mathit{Y}\mathbf{=}\mathit{c}\mathbf{+}\mathit{m}\mathit{x}$, where

  • $Y={\log }_{n}y$ (the logarithm of the output values)

  • $m={\log }_{n}b$ (the slope)

    • this is the linear rate of change on the semi-log plot

  • $c={\log }_{n}a$ (the y-intercept)

    • this is the initial linear value on the semi-log plot

• So on a semi-log plot, the linearized model corresponding to $y=a{b}^x$ is

  • ${\log }_{n}y={\log }_{n}a+\left({\log }_{n}b\right) x$

• E.g., for $y=2·3^x$ (and using base 10 logarithms)

  • ${\log }_{10}y={\log }_{10}2+\left({\log }_{10}3\right)x$

    • or rounding the logs on the right-hand side to 3 decimal places gives

      • ${\log }_{10}y=0.301+0.477x$

  • The slope of the semi-log plot is ${\log }_{10}3\approx 0.477$

    • and the y-intercept is ${\log }_{10}2\approx 0.301$

How can I use the linearization to build an exponential model?

• Because the linearized data follows a straight line

  • you can use linear techniques

    • e.g. finding slope and intercept from two points, or linear regression

  • to model the data on the semi-log plot

• Once you have the slope $m$ and y-intercept $c$ of the line on the semi-log plot

  • The base of the exponential function is $b=n^m$

    • where $n$ is the logarithm base used

    • This follows from $m={\log }_{n}b$

  • The initial value is $a=n^c$

    • This follows from $c={\log }_{n}a$

  • The exponential model is $y=a{b}^x$

• E.g. suppose on a semi-log plot (base 10), data appears linear with slope $0.3$ and y-intercept 1.5

  • Then $b={10}^{0.3}=1.995262...\approx 2$

    • and $a={10}^{1.5}=31.622776...\approx 31.623$

  • The exponential model is $y\approx 31.623·2^x$