  We characterize the Carleson measures for an exponential Bergman space on the
unit ball of $\mathbb C^n$ in terms of the ball induced by the complex Hessian
of the logarithm of the weight function. The boundedness (or compactness) of
integral operators, Ces\`{a}ro operators and Toeplitz operators, is given using
the Carleson measure (or vanishing Carleson measure) characterization.
