Multiplicative bijections of $\mathcal{C(X},I\mathcal{)}$

Let $\mathcal{X}$ be a compact Hausdorff space which satisfies the first axiom of countability, let $I=\left[ 0,1\right]$ and let $\mathcal{C}(\mathcal{X}$,$I)$ be the set of all continuous functions from $\mathcal{X}$ to $I.$ If $\varphi:\mathcal{C}(\mathcal{X}$, $I)\rightarrow\mathcal{C}(\mathcal{X}$,$I)$is a bijective multiplicative map, then there exist a homeomorphism $\mu: \mathcal{X\rightarrow X}$ and a continuous map $k:\mathcal{X} \rightarrow\left( 0,\infty\right) ,$ such that $\varphi (f)(x)=f(\mu(x))^{k(x)}$ for all $x\in\mathcal{X}$ and for all $f\in \mathcal{C}(\mathcal{X},I).$