Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair $(X,\Delta_X)$ is klt. Let us take the image $\Delta_Y=f(\Delta_X)\subset Y$. Is the pair $(Y,\Delta_Y)$ still klt or not?

I am actually interested in the following case. Let $f:\mathbb{P}^n\dashrightarrow\mathbb{P}^n$ be the standard Cremona, and let $p_1,...,p_{n+3}\in \mathbb{P}^n$ be general points. We take $p_1,...,p_{n+1}$ as the center of the Cremona. Let $X$ be the blow-up of $\mathbb{P}^n$ at $p_1,...,p_{n+3}$. Then $f$ induces a small birational map $\overline{f}:X\dashrightarrow X$. If $(X,\Delta)$ is a klt pair then is $(X,\overline{f}(\Delta))$ a klt pair as well ?