2. Among the statements about the function $y=\log _{x}(x+1)$, which one is incorrect () ?
A. Its domain is $\{x \mid x>0$ and $x \neq 1\}$
B. For any real number $a$ other than 0 and 1, $f(x)=a$ has real roots
C. $f(x)$ is decreasing on $(1,+\infty)$
D. $f(x)<f(x+1)$ does not always hold for $x$ in the domain
2. B

A The correctness of this is easily known from the definition of logarithmic operations.
B When $x \in(0,1)$, $f(x)$ $\log _{x} x=1$, so when $a \in(0,1)$, the equation has no solution.
$\mathrm{C}$ Let $x_{1}>x_{2}>1$. Then $\lg \left(1+\frac{1}{x_{1}}\right)f(x+1)$.