Which Produces a Rational Number?

Question

Among the options below, which one produces a rational number when divided by ln(x + 1) for all x > 1?

a) \({ln(x^4-4x^3+6x^2+1)}\)

b) \(ln(x^2-1)\)

c) \(ln(x^2-2x+1)\)

d) \(ln(\sqrt{x-1})\)

e) \(ln(x^3-1)-ln(x^2-x+1)\)

Answer 1

(c) ln(x² - 2x + 1)

 = ln((x - 1)²) = 2 ln(x - 1)

Hence, \(\frac{ln(x^2-2x+1)}{ln(x-1)}=\frac{2ln(x-1)}{ln(x-1)}\) = 2 \(\forall\)x >1

Which is a rational number.

(d) \(\frac{ln\sqrt{x-1}}{ln(x-1)}=\cfrac{\frac12 ln(x-1)}{ln(x-1)}\) = 1/2 a rational number

Comments

The problem statement asked you to divide by ln(x + 1). Why did you divide by ln (x - 1)?