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Answer:
not generally
Explanation:
The basic log function has a vertical asymptote where its argument is zero. It has a positive slope for all positive arguments. It tends to infinity as the argument tends to infinity. There is no horizontal asymptote.
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A log function can be made part of a larger expression in a way that would give the expression a horizontal asymptote. For example, the function ...
f(x) = 1/log(x)
has a horizontal asymptote at f(x) = 0. (This asymptote is approached very slowly.)