If Y is normal, then exp(Y) is lognormal.

Note the support of the Y is the set of real number

but the domain of logarithm function is just the set of positive real number

So log(Y) is not well defined

Both wikipedia and mathworld are correct.

"A continuous distribution in which the logarithm of a variable has a normal distribution."

This sentence means if X ~ lognormal, then lnX ~ Normal