A function y is expected to be of the form y=cx m and the XY data develop a straight line on...

Question:

A function y is expected to be of the form y=cx{eq}^{m} {/eq} and the XY data develop a straight line on log-log paper. The line passes through the (x, y) points (100, 50) and (1000, 10). What are the values of c and m?

Log-log Plot

Log-log plots are very useful when solving a large set of data, where only a few of the data points are much larger than the bulk of the data set. Equations of the form {eq}y = cx^m {/eq} are written as {eq}log(y) = log(c) +mlog(x) {/eq} which is a straight line and makes it easier to analyze.

Answer and Explanation:


The given equation is,

{eq}y = cx^m {/eq}

Representing the above equation on a log-log graph would give us a straight line of the form,

{eq}log(y) = log(c) + m log(x) {/eq}

It is given that this straight line passes through the points (100, 50) and (1000, 10). Substituting these values in the above equation we get a set of simultaneous equations,

{eq}log (50) = log(c) + m log (100) \\ log (10) = log (c) + m log(1000) {/eq}

Subtracting the second equation from the first we get,

{eq}log (50) - log (10) = m*(log(100) - log(1000)) \\ m = -0.69897 {/eq}

Substituting the value of m in the second equation we get,

{eq}log (10) = log (c) - 0.69897*log(1000) \\ log(c) = 3.09691 \\ c = 1250 {/eq}

Therefore we get, m = -0.69897 & c = 1250


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