Is exponential function in a C*-algebra injective on self-adjoint elements?

Question

Let $A$ be a C*-algebra and $\exp(x)=\sum_{n=0}\frac{x^n}{n!}$, the usual exponential function from $A$ into $A$. Is it true that if $x\ne y\in A$, $x^*=x$, $y^*=y$, then $\exp(x)\ne\exp(y)$?