Calculus: The Language of Change

Exercise Set 8.1
The Official Natural Exponential

by  Michelle Nguyen

Exercise 3)  Log Linearity

Show that  if y grows at a constant  percentage  rate  with respect to x, then the quantity  z = Log[y] is a linear  function  of x, z = kx + (Subscript[Z, 0]).  Give  the  value  of  (Subscript[Z, 0])  in  terms  of  (Subscript[Y, 0])= y[0].

Solution

Given:  z = Log[y]

Theorem 8.2 on page 127:  Log[y] = k x + Log [(Subscript[Y, 0])]

So z = k x + Log [(Subscript[Y, 0])]

But Log [(Subscript[Y, 0])] is a constant, which we can call (Subscript[Z, 0]).

Therefore  z = k x +  (Subscript[Z, 0]) and z is a linear function of x with
		(Subscript[Z, 0]) = Log [(Subscript[Y, 0])]