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.
