Lebesgue integral of exponential

Question

Assume that $p>1$ and $e^f\in L^p[0,1]$ and assume that $f_n$ is a sequence of bounded function converging pointwise to $f$ and $\|f_n-f\|_{L^p[0,1]}\to 0$. Can we conclude that $\|e^{f_n}-e^f\|_{L^p[0,1]}\to 0$?