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202202081107 Solution to 1980-AL-PHY-I-33

In an experiment, a student takes several readings of the variable x and y. He plots a graph of \log y against \log x and finds that it is a straight line of gradient m, making an intercept c on the positive \log y axis.

What is the relation between x and y?

Solution.

\begin{aligned} \log y & = m\log x+c\\ \log y & = \log x^m + c\\ \log y-\log x^m & = c\\ \log \bigg( \frac{y}{x^m}\bigg) & = c \\ \cdots\enspace \textrm{let }c =\log a&\enspace \cdots\\ \log \bigg( \frac{y}{x^m}\bigg) & = \log a \\ \frac{y}{x^m} & = a\\ y & = ax^m \\ \end{aligned}