I’m trying to understand how to linearize functions using logs and I can’t quite wrap my head around this.
Let’s say we have two functions:
p(x) = x^2
q(x) = x(i-1)*1.5 (for x >= 1, q(1) = 1)
p:
p(1) = 1
p(2) = 4
p(3) = 9
…
Linearize p (if you graph the coordinates below, you get a straight line):
log(1), log(1)
log(2), log(4)
log(3), log(9)
q:
q(1) = 1
q(2) = 1.5
q(3) = 2.25
…
Linearize q:
1, log(1)
2, log(2)
3, log(3)
To linearize p, we have to take the log of both x, and p to make the function linear. To linearize q, we have to take the log of just q to make the function linear.
What is the rule for this, or how can I think through this to determine how to linearize a function without trial and error?