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For each unital $C^*$-algebra $A$, we denote $cel_{CU}(A)=\sup\{cel(u):u\inCU(A)\}$, where $cel(u)$ is the exponential length of $u$ and $CU(A)$ is theclosure of the commutator subgroup of $U_0(A)$. In this paper, we prove that$cel_{CU}(A)=2\pi$ provided that $A$ is an $AH$ algebras with slow dimensiongrowth whose real rank is not zero. On the other hand, we prove that$cel_{CU}(A)\leq 2\pi$ when $A$ is an $AH$ algebra with ideal property and ofno dimension growth (if we further assume $A$ is not of real rank zero, we have$cel_{CU}(A)= 2\pi$).

$C^*$ exponential length of commutators unitaries in $AH$ algebras