Exponential law and semi-log graph

An exponential law has the form

$y = a\cdot k^{x}, \qquad k > 0,\ k \neq 1$

and describes situations in which, for every unit increase in $x$, the quantity $y$ is multiplied by a fixed factor $k$. If $k > 1$ the quantity grows (compound interest, a bacterial population); if $k < 1$ it decreases, losing a fixed percentage of its current value at every step (radioactive decay, the bounce heights of a ball). For example $k = 0{,}87$ means that at every step $y$ loses $13\%$ of its value at that moment.

Linearisation: why the semi-log works

The semi-logarithmic graph has the $y$ axis on a logarithmic scale and the $x$ axis on a linear scale. The reason it straightens out exponentials can be seen by taking the $\log_{10}$ of both sides and using the properties of logarithms (product → sum, power → multiplication):

$\underbrace{\log_{10}(y)}_{Y} = \underbrace{\log_{10}(a)}_{q} + x\cdot\underbrace{\log_{10}(k)}_{m}$

In the plane $(x,\,\log_{10} y)$ this is a straight line $Y = q + mx$: the exponential law, which was a curve on a linear graph, becomes a straight line. If the experimental points line up on a semi-log graph, then the law is exponential — and vice versa.

Recovering the parameters from a semi-log graph

$m = \frac{\Delta Y}{\Delta x} \;\Rightarrow\; k = 10^{m} \qquad\qquad q = Y\big|_{x=0} \;\Rightarrow\; a = 10^{q}$

The sign of the slope immediately tells you what type of exponential it is: if $m < 0$ then $k < 1$ and it is a decay; if $m > 0$ then $k > 1$ and it is a growth. The slope measures the rate of change per step, while the intercept measures the starting value $a = 10^{q}$.

The typical application is the bouncing ball experiment: the maximum height $h$ is measured at each bounce, $h = a\cdot k^{n}$ is expected with $k < 1$, and from the best-fit line on the semi-log graph both the fraction of energy lost ($k$) and the starting height ($a$) can be read off.

Links

Topics: Experimental method Skills: Use of logarithmic scales · Data linearisation · Reading graphs

Related exercises: Worked exercise — Bouncing ball (semilog) · Problem — Bacteria and semilog graph · Problem — Fermi estimate, measurements of g