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This correspondence shows that the support growth of a fixed-rate optimum (minimum mean-squared error) scalar quantizer for a Laplacian density is logarithmic with the number of quantization points. Specifically, it is shown that, for a unit-variance Laplacian density, the ratio of the support-determining threshold of an optimum quantizer to 3/√2lnN/2 converges to 1, as the number N of quantization points grows. Also derived is a limiting upper bound that says that the support-determining threshold cannot exceed the logarithmic growth by more than a small constant, e.g., 0.0669. These results confirm the logarithmic growth of the optimum support that has previously been derived heuristically.