Logarithmic graphs can be used to estimate parameters in relationships of the form:

\(y=ax^n \) and \(y=kb^x \)

given data for \(x\) and \(y\).

\(\bm{\underline{y=ax^n}} \)

\(y=ax^n \quad \) (\(\log\) both sides)

\(\implies \log{y}=\log{ax^n} \quad \) (apply addition law)

\(\implies \log{y}=\log{a} + \log{x^n} \quad \) (apply power law)

\(\implies \log{y}=\log{a} + n\log{x} \quad \) (compare this with \(y = mx + c\))

To estimate \(a\) and \(n\) in \(y=ax^n \), plot \(\log{y}\) against \(\log{x}\) and obtain a straight line where the \(y\)-intercept is \(\log{a}\) and the gradient is \(n\).

\(\bm{\underline{y=kb^x}} \)

\(y=kb^x \quad \) (\(\log\) both sides)

\(\implies \log{y}=\log{kb^x} \quad \) (\(\log\) both sides)

\(\implies \log{y}=\log{k} + \log{b^x} \quad \) (apply addition law)

\(\implies \log{y}=\log{k} + x\log{b} \quad \) (compare this with \(y = mx + c\))

To estimate \(k\) and \(b\) in \(y=kb^x \), plot \(\log{y}\) against \(\log{x}\) and obtain a straight line where the \(y\)-intercept is \(\log{k}\) and the gradient is \(\log{b}\).