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Data from an experiment may result in a graph indicating exponential growth. This implies the formula of this growth is \(y = k{x^n}\), where \(k\) and \(n\) are constants.

Using logarithms, we can express \(y = k{x^n}\) in the form of the equation of a straight line \(y = mx + c\).

Shown below is a straight line graph when \({\log _{10}}y\) is plotted against \({\log _{10}}x\). The straight line passes through \((0,6)\). Express \(y\) in terms of \(x\).

As it shows the graph of a straight line, we begin with the equation \(y = mx + c\). From the graph, we can also see that the y-intercept is 6, therefore we can say that the equation of the straight line is \(y = mx + 6\).

\(m = \frac{{vertical}}{{horizontal}}\)

\(m = \frac{6}{2} = 3\)

So \(y = 3x + 6\). However, instead of an \(x\) and \(y\) axis, we have \({\log _{10}}y\) and \({\log _{10}}x\) axes. Therefore:

\(y = 3x + 6\)

\({\log _{10}}y = 3{\log _{10}}x + 6\)

Using \({a^x} = y\) and \({\log _a}y = x\), we can change the '6' into a log.

\({\log _{10}}y = {\log _{10}}{x^3} + {\log _{10}}{10^6}\)

\({\log _{10}}y = {\log _{10}}{10^6}{x^3}\)

\(y = 1000000{x^3}\)