In 1760, Daniel Bernoulli applied exponential and logarithmic reasoning to model the impact of smallpox inoculation on life expectancy. He assumed that the risk of smallpox infection was constant over time, leading to an exponential survival function, meaning the probability of not getting smallpox declined exponentially with age. His goal was to show mathematically that widespread inoculation could reduce deaths, even if it carried some risks. This early use of exponential functions helped lay the groundwork for modern epidemic modeling, showing students how math can literally save lives.