I have f'(x) = x(2ln(x) - 3) / (ln(x) - 1)^2 (which matches my book's answer)

My book says setting this equal to zero gives:

2ln(x) - 3 = 0

I think this implies that x / (ln(x) - 1)^2 equals 1
or (ln(x) - 1)^2 = x
but I don't understand why!

I need to see the steps.

Thanks,

if f'(x) is set to zero.. it implies x=o or 2ln(x)-3 is zero. Like saying A.B = zero means A is zero or B is zero. In this case A is x and 2ln(x)-3 is B. In this case we have A.B/C we are concerned with terms that make it zero.. obviously A or B being zero.

I have f'(x) = x(2ln(x) - 3) / (ln(x) - 1)^2 (which matches my book's answer)


I think this implies that x / (ln(x) - 1)^2 equals 1
or (ln(x) - 1)^2 = x
but I don't understand why!



Sorry, but if 2 ln(x) - 3 = 0, this does not imply that x / (ln(x) - 1)^2 equals 1. Why do you think it does?

If you say A.B / C = 0
Implies A= 0 or B = 0 But C must not equal to zero.


f'(x) = x(2ln(x) - 3) / (ln(x) - 1)^2 =0
That means x = 0 or 2 ln(x) - 3 = 0.
When x = 0, the denominator (ln(x) - 1)^2 also equal to 0.
It forms a 0/0, it is undefined.
That means x can't be zero.

So the only value makes f'(x) zero is 2 ln(x) - 3 = 0.

if a/b=o it implies a=0 but this (a=0)doesn't mean b=1 or 1/b=1 .so b can be any number except 0.the value of b doesn't affect the equation.cause a=0*b=0 for any value of b.