Q) Find the domain of the defintion of the function :-
Y = log(1 - log(x^2 - 5x + 16))
Any help would be greatly appreciated.
Thanks,
Ashish
Since the log-function is defined for positive real numbers only
$\displaystyle 1 - \log(x^2 - 5x + 16)>0~\implies~\log(x^2 - 5x + 16)<1$ ....and $\displaystyle \log(x^2 - 5x + 16)>0$
De-logarithmize(?) both sides:
$\displaystyle x^2-5x+6<10~\implies~x^2+5x+6<0$
The LHS can be factored:
$\displaystyle (x+1)(x-6)<0$ ......... Aproduct of 2 factors is negative (< 0) if one factor is negative and the other factor is positive:
$\displaystyle \begin{array}{rcl}x+1<0\wedge x-6>0 & ~\vee~ & x+1>0\wedge x-6<0 \\ x<-1\wedge x>6 &~\vee~& x>-1\wedge x<6 \\ \emptyset &~\vee~&\boxed{-1<x<6}\end{array}$
2nd condition:
$\displaystyle \log(x^2 - 5x + 16)>0~\implies~x^2-5x+16>1$
$\displaystyle x^2-5x+15>0~\implies~x\in \mathbb{R}$
Therefore the domain of the function is $\displaystyle d=(-1,6)$