Question

If $f\left(x\right)=\log (x-2)+\log (x-3)$ and $\varphi \left(x\right)=\log (x-2)(x-3).$
Are both functions equal?

Answer 1

For $f\left(x\right)$ to be defined, $x-2>0$ and $x-3>0$

So domain of $f\left(x\right)$ is $(3,\infty )$

And for $\varphi \left(x\right)$ to be defined $(x-2)(x-3)>0$

$\Rightarrow x\in (-\infty ,2)\cup (3,\infty )=$ domain of $\varphi \left(x\right)$

But we know that two function are same, if their range ,domain and values at each $x$ is equal

Hence here $f$ and $\varphi$ are not equal