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Add random numbers, get normal what occurs as multiply

When you add random numbers and get normal, what occurs when you multiply them?

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In orders of logarithms of the random numbers, logarithms of random numbers are themselves random (assume that stay with logarithms of strictly positive numbers). Therefore if you add up many logarithms of random numbers you will find out a normal distribution. But, obviously, a sum of logarithms is only the logarithm of a product, thus the logarithm of the product should be normal, and it is the definition of lognormal: the product of positive random numbers converges to lognormal.

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