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## GR0877 Problem 13

PROBLEM STATEMENT: This problem is still being typed.

SOLUTION: (A) On a log-log plot a function $y = ax^m$ will appear as a straight line, where $m$ is the slope of the line and $a$ is the $y$-value corresponding to $x = 1$. Indeed, $\log_{10}{y} = \log_{10}(ax^m) = \log_{10} a + m \log_{10} x$. Thus if $x = 1$, then $y = a$ and $\displaystyle m = \frac{\log_{10} y_2 - \log_{10} y_1}{\log_{10} x_2 - \log_{10} x_1} = \frac{\log_{10} (y_2/y_1)}{\log_{10} (x_2/x_1)}$. In our case, for $x = 1$, $y \approx 6$ and $\displaystyle m = \frac{\log_{10} (100/10)}{\log_{10}(300/3)} = \frac{1}{2}$. Therefore, $y = 6 \sqrt{x}$.

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P.S. Actually, you don’t have to do this at all if you found $m$. The only choice that have a form $\sim \sqrt{x}$ is (A). The extrapolation was made in order to find the corresponding coefficient $a$ in the function $y = a x^m$.