 Hello and welcome to the session. In this session we will discuss a question which says that suppose that 500 milligrams of a medicine enters a patient's bloodstream at noon and decays exponentially, the exponential function d of t is equal to 500 into 10 raised to power minus 0.07 t models the situation where t is time in hours and d of t is the amount of medicine active in the patient's body find the time when only 5% of the original amount of medicine will be active in patient's body. Now let us start with the solution of the given question. Now in this question we are given that 500 milligrams of a medicine enters a patient's body at noon and decays exponentially and the exponential function that represents the given situation is d of t is equal to 500 into 10 raised to power minus 0.07 t where t is time in hours and d of t is the amount of medicine active in the patient's body. Now the given exponential function is of the type a into b raised to power ct which is equal to d where b is the base. Now in this exponential function base b is equal to 10 and we have to find the time t when only 5% of the original amount of medicine will be active in patient's body. So we have to find time t when d of t is equal to 5% of original amount of medicine. Now original amount of medicine is 500 milligrams. So we have to find time t when d of t is equal to 5% of 500 which is equal to 5 upon 100 into 500 and this is equal to 25. So we have to find time t when d of t is equal to 25. For this we will put d of t is equal to 25 and the given exponential function supporting d of t is equal to 25 and this exponential function we have 25 is equal to 500 into 10 raised to power minus 0.07 t. Now dividing both sides by 500 we have 25 upon 500 is equal to 500 upon 500 into 10 raised to power minus 0.07 t. This further implies now 25 into 20 is 500. So here it would be 1 upon 20 is equal to 10 raised to power minus 0.07 t. Then here you can see we have base 10. So it will be convenient if we take logarithms on both sides with base 10. So taking common logarithm on both sides we have log 1 upon 20 to the base 10 is equal to log 10 raised to power minus 0.07 t to the base 10. Now we can simply write log n to the base 10 as log n. So here we can write log 1 upon 20 is equal to log 10 raised to power minus 0.07 t. Now we know that log m over n is equal to log m minus log n. So here log 1 upon 20 is equal to log 1 minus log 20 and this is equal to now again we know log n raised to power n is equal to n log m. So here log 10 raised to power minus 0.07 t is equal to minus 0.07 t into log 10 and we also know that log 10 is equal to 1 and log 1 is equal to 0. So this implies 0 minus log 20 is equal to minus 0.07 t into log 10. Now log 10 is 1 so minus 0.07 t into 1 is equal to minus 0.07 t and this implies minus log 20 is equal to minus 0.07 t and this further implies log 20 is equal to 0.07 t. Now using calculator log 20 is equal to 1.301 so this implies 1.301 is equal to 0.07 t. Now dividing both sides by 0.07 we have 1.301 upon 0.07 is equal to 0.07 upon 0.07 into t. Now further on solving this implies 18.58 is equal to t and this implies t is approximately equal to 18.6 thus 25 milligrams of medicine will be active in patient study after 18.6 hours. So this is the solution of the given question and that's all for this session. Hope you all have enjoyed the session.