Reducing to straight line Example Two variables a and b are related by the formula s=atc, where a and c are constants Show that this relationship can be written as : Log s = c logt + loga S = atc Log S = Log atc Log S = Log a + Log tc (using the log law loga xy= loga x + loga y) Log S = Log a + c Log t (using the log law loga xk= k loga x ) Log S = c Log t + Log a

(ii) Explain why the model can be tested by plotting log y vs log x Compare Log S = c Log t + Log a With the equation of straight line y = m x + c Log S = c Log t + Log a should give a straight line if s=atc where : y = log S x = log t m(gradient) = c c(intercept) = Log a

(iii) Plot log s vs log t and estimate the values of a and c 9 13 16 18 20 22 t 5 10 15 25 30 log s 0.954 1.114 1.204 1.255 1.301 1.342 log t 0.699 1.0 1.176 1.398 1.477

log s 0.954 1.114 1.204 1.255 1.301 1.342 log t 0.699 1.0 1.176 1.398 1.477 This looks very linear to me But I need a carefully drawn graph that I can estimate the values of the gradient and intercept from. So I will use Excel

Log s = c logt + loga C (gradient) = Log a S=atc is data relationship where a = 3.98 and c =0.5 Log a = 0.6  a= 100.6 = 3.98 C = 0.5