 Hello and welcome to the session. In this session we will discuss the solution of exponential models using logarithms. Now we have an exponential model of type a b raised to the power c t which is equal to b where b is the base. Now b can be 10 it can be e or any number other than 10 or e. Now to solve this model using logarithms we can take either common logarithms or natural logarithms. Now when base is 10 or other than 10 we use common logarithms that when base is e it is always convenient to take natural logarithms. Now let us discuss an example. Now suppose the number of bacteria in a lab dish at any time t r's after an experiment is given by the function b of t is equal to 50 into 10 raised to the power 0.3 t. Now we want to find the time 3 when there will be 20,000 bacteria in the lab dish. Now this is an exponential model of form a b raised to the power c t which is equal to d where b is base. Now the given function is b of t is equal to 50 into 10 raised to the power 0.3 t. Now here you can see base b is 10 and we know that when base is 10 then we use common logarithms. So solving this exponential model using common logarithms we have b of t is equal to 50 into 10 raised to the power 0.3 t and we have to find the time t when there will be 20,000 bacteria in lab dish. So we are given b of t is equal to 20,000 and we have to find time t. Now let us put b of t that is 20,000 in the given model. So we have 20,000 is equal to 50 into 10 raised to the power 0.3 t. Now dividing both sides of this equation by 50 we have 20,000 upon 50 is equal to 50 upon 50 into 10 raised to the power 0.3 t. Now we know that 50 into 400 is 20,000. So this implies 400 is equal to 10 raised to the power 0.3 t. Now since the base is 10 so it is convenient to take common logarithms on both sides with base 10. So taking logarithms on both sides we get log 400 is equal to log 10 raised to the power 0.3 t. Now using the property of logarithm we know that log n raised to the power n is equal to n log n. So here we have log 400 is equal to, now here using this property we have 0.3 t log 10 that is log 10 raised to the power 0.3 t is equal to 0.3 t log 10. Also we know log 10 to the base 10 is equal to 1 or simply we can say log 10 is equal to 1. So this implies log 400 is equal to 0.3 t into log 10. Now log 10 is 1 so 0.3 t into 1 is equal to 0.3 t. Now using calculator we have log 400 is equal to 2.602. So this implies 2.602 is equal to 0.3 t Now dividing both sides by 0.3 we have 2.602 upon 0.3 is equal to 0.3 t upon 0.3. Now solving this implies 6.7 is equal to t. Thus in 8.67 hours the number of bacteria in the lab dish will be 20,000. Now let us discuss how can we use logarithms to solve exponential model when base is, now let us see one example. Now here it is given the atmospheric pressure p in pounds per square inch at x minus above c level is given approximately by p is equal to 14.7 into e raised to the power minus 0.2 where x. Now here we have to find height x when atmospheric pressure p is equal to 7.35. Now putting e is equal to 7.35 in the given model we have 7.35 is equal to 14.7 into e raised to the power minus 0.21x. Now dividing both sides by 14.7 this implies 7.35 upon 14.7 is equal to 14.7 upon 14.7 into e raised to the power minus 0.21x. Now solving this implies 0.5 is equal to raised to the power minus 0.21x. Now here base is e. So it is convenient to take natural logarithm on both sides of this equation. So we have natural log of 0.5 is equal to natural log of e raised to the power minus 0.21x. Now we know that natural log of e raised to the power is equal to a. So we have natural log of 0.5 is equal to now here natural log of e raised to the power minus 0.21x will be minus 0.21x. Now dividing both sides by minus 0.21 this implies natural log of 0.5 upon minus 0.21 is equal to minus 0.21 upon minus 0.21 into x. Now using scientific calculator we have natural log of 0.5 as minus 0.69314718 upon minus 0.21 is equal to x. Now simplifying this implies 3.3 is equal to x thus x is equal to 3.3 miles. So at high 3.3 miles the atmospheric pressure will be 7.35. Now let us consider another example when we have base other than 10 or e here. Suppose we have exponential model given by b of t is equal to 50 into 2 raised to power 0.3t. Now we want to find t when b of t is equal to 60,000. Now in this exponential model you can see base b is equal to 2. Now let us put b of t is equal to 60,000 in the given exponential model. So here we have 60,000 is equal to 50 into 2 raised to power 0.3t. Now dividing both sides by 50 we have 60,000 upon 50 is equal to 50 upon 50 into 2 raised to power 0.3t. Now solving we have 1200 is equal to 2 raised to power 0.3t. Now we take common logarithm on both sides of this equation and we have log 1200 is equal to log 2 raised to power 0.3t. Now using this property of logarithm we have log 1200 is equal to 0.3t into log 2. Now we write both sides of this equation by log 2. So here we have log 1200 upon log 2 is equal to 0.3t into log 2 upon log 2. This implies log 1200 upon log 2 is equal to 0.3t. Now using calculator we have got the values of log 1200 and log 2. So putting these values in this equation we have 3.079 upon 0.301 is equal to 0.3t. Now solving this implies 10.23 is equal to 0.3t. Now dividing both sides by 0.3 we have 10.23 upon 0.3 is equal to 0.3 upon 0.3 into t which further implies 34.1 is equal to t thus t is equal to 34.1 when v of t is equal to 60,000. So in this session we have discussed the solution of exponential models using logarithms and this completes our session. Hope you all have enjoyed the session.