Problem — True or false about graphs

Problem

True or false, with justification: (a) if on a $y$–$x$ graph the points line up, the law is $y = mx + q$; (b) on a semi-log graph with $y$ on a logarithmic scale, a straight line corresponds to a power law $y = a\,x^n$; (c) a log-log graph allows the exponent to be read as the slope.

Solution

(a) TRUE. If, on a graph with both axes linear, the experimental points lie along a straight line, the relationship between the two quantities is linear, that is $y = mx + q$, where $m$ is the slope and $q$ the intercept. This is precisely the geometric definition of a linear law: alignment on linear axes.

(b) FALSE. On a semi-logarithmic graph only the ordinate is on a logarithmic scale, while the abscissa stays linear. If the points line up it means $\log y$ is linear in $x$, that is $\log y = \log a + x\log k$, giving $y = a\,k^{x}$: an exponential law, not a power law. A power law $y = a\,x^{n}$ gives a straight line only when both axes are logarithmic (log-log), because there $\log y = \log a + n\log x$.

(c) TRUE. On a log-log graph a power law $y = a\,x^{n}$ becomes $\log y = \log a + n\log x$: it is a straight line whose slope is exactly the exponent $n$. Measuring the slope of the line therefore directly gives the exponent of the power law (and $a$ can be obtained from the intercept).

$\ev{\text{(a) V} \qquad \text{(b) F} \qquad \text{(c) V}}$

Links

Topics: Experimental method Skills: Reading graphs · Using logarithmic scales