Solution assignment 04 Logarithmic functions and graphs

Under which conditions is the following logarithmic function defined?

$\log_{c+2}(x^2-3x+2)$

First the base $c+2$ has to satisfy:

$c+2>0$ and $c+2\neq1$

so:

$c>-2$ and $c\neq-1$

Further the argument has to satisfy:

$x^2-3x+2>0$

The graph of the function at the left-hand side is an 'opens up' parabola and the function can be factorized:

$(x-2)(x-1)>0$

The parabola lies above the $X$ -axis if:

$x<1$ or $x>2$

and for these values of $x$ the logarithm is defined.