When to use semilog and when to use log-log

The three cases you meet in the lab can be summarised in a single strategy: transform the data so the points line up, then read the parameters off the best-fit line. What changes is which transformation is needed.

Three laws, three graphs

Type of law	Graph	From the fitted line
$y = a\cdot k^{x}$ (exponential)	Semilog ($y$ on log scale, $x$ linear)	$m = \log k \Rightarrow k = 10^{m}$; $q = \log a \Rightarrow a = 10^{q}$
$y = a\cdot x^{n}$ (power)	Log-log (both axes on log scale)	$m = n$ (the exponent); $q = \log a \Rightarrow a = 10^{q}$
$y = mx + q$ (linear)	Normal scale	$m$ = slope; $q$ = intercept

The practical rule for choosing: if you suspect the variable sits in the exponent (something that doubles or halves at each fixed step), try semilog. If you suspect the variable sits at the base raised to a power (something that scales like a square, a root, an inverse square), try log-log. In both cases the verdict is visual and immediate: if, after the transformation, the points lie on a straight line, you have guessed the form of the law correctly, and the line itself hands you the parameters.

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Topics: Metodo sperimentale Skills: Uso delle scale logaritmiche · Linearizzazione dei dati

Related exercises: Esercizio svolto — Pallina rimbalzante (semilog) · Problema — Batteri e grafico semilog · Problema — Stima di Fermi, misure di g