Numerical Methods y axb Consider a nonlinear transformation

Numerical Methods

y = ax^b Consider a nonlinear transformation of the above equation. Linear regression of In x and in y resulted in the following equation: y = 0.192x -0.335 What are the values of a and b? A) a = 0.715 b = 0.1922 B) a = -0.335 b = 0.192 C) a = -0.3345 b = 1.212 D) a = 0.967 b = 0.1922

Solution

Given equation,

y = ax^b

Taking ln on both sides,

ln(y) = ln(ax^b)

By multiplication rule of logarithms,

ln(y) = ln(a) + ln(x^b)

By power rule of logarithms,

ln(y) = ln(a) + bln(x) ----------------- (1)

It is said that linear regression of ln(x) and ln(y) resulted in

y = 0.192x - 0.335

Using actual variables the equation will be,

ln(y) = 0.192ln(x) - 0.335 ------------------------ (2)

Comparing equations (1) and (2),

ln(a) = -0.335

b = 0.192

a = 0.715

So the correct option is (a).