Single-molecule measurements provide a platform for investigating the dynamical properties of enzymatic reactions. To this end, the single-molecule Michaelis-Menten equation was instrumental as it asserts that the first moment of the enzymatic turnover time depends linearly on the reciprocal of the substrate concentration. This, in turn, provides robust and convenient means to determine the maximal turnover rate and the Michaelis-Menten constant. Yet, the information provided by these parameters is incomplete and does not allow access to key observables such as the lifetime of the enzyme-substrate complex, the rate of substrate-enzyme binding, and the probability of successful product formation. Here we show that these quantities and others can be inferred via a set of high-order Michaelis-Menten equations that we derive. These equations capture universal linear relations between the reciprocal of the substrate concentration and distinguished combinations of turnover time moments, essentially generalizing the Michaelis-Menten equation to moments of any order. We demonstrate how key observables such as the lifetime of the enzyme-substrate complex, the rate of substrate-enzyme binding, and the probability of successful product formation, can all be inferred using these high-order Michaelis-Menten equations. We test our inference procedure to show that it is robust, producing accurate results with only several thousand turnover events per substrate concentration.