Domain is the set of x-values that will define or validate f(x), or make f(x) true.

ln is natural log.

(-infinity,0] don't define logs, so x cannot be from -infinity up to zero.

(0,infinity) will define logs, so x can be from, but not including, 0 to infinity.

f(x) = -ln(1/x)

f(x) = (-1)ln(1/x)

f(x) = ln[(1/x)^(-1)]

f(x) = ln(1 / 1/x)

f(x) = ln(x)

That means x can be from right after zero up to right before infinity.

Therefore, domain of f(x) is (0,infinity). ----answer.