Do you mean, you have some function, f, and its logarithm, ln(f), and you want to find value of x that minimizes both f and ln(f)?
Assuming there are no "boundaries" then we are looking for critical points, where the derivative is 0 or does not exist. The derivative of f is f' and the derivative of ln(f) is f'/f. A fraction is 0 if and only if its numerator is 0. Assuming that f(x) is not 0, in which case ln(f(x)) would not exist, f'(x)/f(x) is 0 if and only if f'(x) is 0. Further, since ln(x) is an increasing function, a maximum for f(x) implies a maximum for ln(f(x)) and vice-versa.l
