# Is there some reason why $\ln(x^2+x) - \ln x = \ln (x+1)$ would not always be true?

I ask because although I've learned in calculus class that it is a valid algebraic rule to work with logarithms, wolfram alpha doesn't unequivocally say that it is always true, which leaves me wondering if there is some condition I'm missing somewhere.

Thank you for any help!

• Check for $x \in (-1,0]$. In that interval $\ln(x+1)$ is defined but not $\ln(x)$. However your equality is true for all $x>0$.
– Axel
Jun 24, 2021 at 8:17
• It's true where everything is defined, but it's not always defined Jun 24, 2021 at 8:19

The domain of LHS is $$(0,\infty)$$ while that of RHS is $$(-1,\infty)$$
So yes , LHS=RHS would not be true in the case of $$x\in (-1,0]$$ because if $$x\in(-1,0]$$ then LHS will be undefined and RHS will be defined and since they are equal it will lead to a contradiction.
They both will be equal to each other in their common domain i.e. $$x\in(0,\infty)$$