L uteinizing hormone stimulates L eydig cells to make testosterone. (FA2020 p628)
Primary gonadal dysfunction (general hypogonadism)would result in low testosterone but high luteinizing hormone because less testosterone is available to cause feedback inhibition in the pituitary. This also makes sense because if the cells are not making enough testosterone but the pituitary thinks they should be, it will keep trying to tell them to make testosterone by secreting more and more luteinizing hormone (LH).
Central dysfunction (dysfunction at the level of the pituitary/hypothalmus or hypogonadotropic hypogonadism) would have low LH because either LH cannot be secreted or there is an issue with understanding the feedback of low testosterone.
Thus measuring LH will let us know where our patient's dysfunction lies.