Power-law model from a log-log graph

Statement

Two quantities $x$ and $y$ are related by $y=A\,x^{n}$. The graph of $\ln y$ against $\ln x$ is a straight line through the points $(0,\ 0{.}7)$ and $(3,\ 3{.}4)$. Find $A$ and $n$ to two decimal places.

Solution

Taking logarithms: $\ln y=\ln A+n\ln x$, a straight line with slope $n$ and intercept $\ln A$.

Intercept: $\ln A=0{.}7\Rightarrow A=e^{0{.}7}\approx 2{.}01$.

Slope: $n=\dfrac{3{.}4-0{.}7}{3-0}=\dfrac{2{.}7}{3}=0{.}90$.

$\boxed{A\approx 2{.}01;\qquad n=0{.}90}$