If the data appear to fit a power function, what might be the most appropriate non-linear transformation, to linearize the data?

If `y` is a power function of `x`, then `y=ax^{n}` for some constants `a` and `n`. Taking the log of both sides (say, the common logarithm (base 10), but any log will do) gives

`log(y)=log(ax^{n})`

Using properties of logarithms, this can be written as

`log(y)=log(a)+nlog(x)`

Letting `Y=log(y)`, `X=log(x)`, and `A=log(a)`, this equation becomes

`Y=A+nX`, giving `Y=log(y)` as a linear function of `X=log(x)`.

For example, suppose your data consisted of the points `(2,6.7)`, `(3,18.8)`, `(4,38.4)`, and `(5,66.9)`. Plotting these data gives a definite nonlinear trend in the graph shown below.

Suppose you suspect the relation between `x` and `y` is a power function. Take the log of both the `x`- and `y`-coordinates of your data, to get `X`- and `Y`-coordinates of data for a log-log plot: `(0.301,0.826),(0.477,1.274),(0.602,1.584),(0.699,1.825)`. This plot has a definite linear trend.

In fact, if you find the least-squares linear regression line for this second graph, you'll get approximately `Y=0.072499+2.5106X`. This implies that `log(y)=0.072499+2.5106log(x)` so that `y=10^{0.072499+2.5106log(x)}=10^{0.072499}*10^{log(x^{2.5106})}\approx 1.18168x^{2.5106}`. The final graph shows that this is a good fit for the original `xy`-data.