Problem — Bacteria and the semilog graph

Problem

You have a table $(t,\ N(t))$ in which the number $N$ of bacteria doubles every hour. Sketch the graph $(t, N)$ on a linear scale and on a semilog scale (with $\log N$ on the vertical axis). Explain, without calculations, why the second case gives a straight line and how to read the doubling time from the slope.

Solution

Growth law. If the number doubles every time $T_2$ (here $T_2 = 1$ h), the population follows an exponential law $N(t) = N_0 \cdot 2^{\,t/T_2}$

Linear scale. With both axes linear, exponential growth appears as a curve that rises ever more steeply: it grows slowly at first, then “explodes”. The shape is concave upward.

Semilog scale. Plotting $\log N$ on the vertical axis (logarithmic) and leaving $t$ linear, we take the logarithm of the law: $\log N = \log N_0 + \frac{t}{T_2}\,\log 2$ This is the equation of a straight line in the variables $t$ and $\log N$: the intercept is $\log N_0$ and the slope is $m = \dfrac{\log 2}{T_2}$ (using $\log_{10}$, $m = \log_{10} 2 / T_2$). The reason a straight line results is that the logarithm turns repeated multiplication (successive doublings) into constant addition: every hour $\log N$ increases by the same amount $\log 2$, and constant increments over equal time intervals give a straight line.

Reading the doubling time. Once the slope $m$ of the line on the semilog graph is measured, the doubling time is obtained by inverting the relation: $T_2 = \frac{\log_{10} 2}{m}$

Sketches. On the left the linear scale (exponential curve), on the right the semilog scale (straight line):

$\ev{\log N = \log N_0 + \dfrac{\log 2}{T_2}\,t \qquad\Rightarrow\qquad T_2 = \dfrac{\log_{10} 2}{m}}$

Links

Topics: Metodo sperimentale Skills: Uso delle scale logaritmiche · Lettura dei grafici