Question

Solve the following inequality:
${\log }_{x-3}(x-1)<2.$

Answer 1

${\log }_{(x-3)}(x-1)<2$

$\Rightarrow \frac{\log (x-1)}{\log (x-3)}<2$

$\Rightarrow \log (x-1)<2\log (x-3)$

$\Rightarrow \log (x-1)<\log (x-3{)}^2$

Remove $\log$ from both the sides:-

$\Rightarrow (x-1)<(x-3{)}^2$

$\Rightarrow x-1<x^2+9-6x$

$\Rightarrow x^2-7x+10>0$

$\Rightarrow x^2-5x-2x+10>0$

$\Rightarrow (x-5)(x-2)>0$

$x-5>0\Rightarrow x>5$ or $x-5<0\Rightarrow x<5$

$x-2>0\Rightarrow x>2$ or $x-2<0\Rightarrow x<2$

$\left[\right(x-5\left)\right(x-2)$ will be greater than zero if and only if $(x-5)$ and $(x-2)$ both are either positive or both are either negative]

$x-5>0$ and $x-2>0$, together will be positive only if $x>5$.

and,

$x-5<0$ and $x-2<0$, together will be negative only if $x<2$

Solution: $x<2$ or $x>5$ but the function becomes undefined in the interval $x<2$.

Interval Notation: $(5,\infty )$.