 Hello and welcome to the session. In this session we will discuss how to find exponential growth and detail for the function. Now we already know about the type of expressions involving paths such as x raise to power 6 and x raise to power 5. Here you can see that the path or exponent is a constant and this is a variable and now we will see exponential expressions. Now the expressions of the form 2 raise to power x, 3 raise to power y, etc. are called exponential expressions where you can see that this is a constant and our exponent is a variable. So such type of expressions where the variable occurs in the exponent are called exponential expressions. Now exponential expressions arise when we have a quantity that changes by the same factor for each unit of time. Now let us discuss an example. Now suppose population of a city doubles every year. Now we have to write an expression when initial population is 3 million. Now we have initial population 3 million. Now since population of a city doubles every year, so after one year population will be 2 into 3 million or we can write it as 2 raise to power 1 into 3 million. Similarly after two years population will be 2 raise to power 2 into 3 million then after three years population will be 2 raise to power 3 into 3 million thus after three years population will be 2 raise to power 3 into 3 million. Thus expression for the growth in population is 3 into 2 raise to power 3 and this is an exponential expression thus an exponential function is a function that can be described by an equation of the form y is equal to a into b raise to power x where b is greater than 0 and b is not equal to 1 also a and b are constants and here a is the initial value or starting value b is the growth factor the amount by which the value of expression gets multiplied for each unit increase in x b is also called base of the exponential expression. The exponential functions can also describe quantities which are decreasing or decaying Now when b is greater than 1 then the expression describes increase or growth of the quantity at a particular rate and when b is less than 1 then the expression describes decrease or decay in the quantity at a particular rate Now let us discuss growth and decay factor and percentages that is growth and decay rate Now suppose we are given that a quantity grows by 6% every year here 6% is the growth rate Now suppose we are given that a quantity decays by 20% every year then 20% will be decay rate of the quantity First of all let us find growth factor let initial value of quantity be a Now given that growth rate is 6% so amount after 1 year will be equal to a plus 6% of a that is equal to a plus 6 upon 100 into a Now from both these terms let us take a common so it will be a into 1 plus 6 upon 100 the whole further this is equal to a into 1 plus 0.06 the whole further this is equal to a into now 1 plus 0.06 is 1.06 So amount after 1 year is equal to a into 1.06 thus growth factor is 1.06 Now after 3 years amount will be a into 1.06 raised to power t Now here b is 1.06 which is greater than 1 is increased Similarly we can find decay factor when decay rate is 20% every year Now let initial amount be a so amount after 1 year will be equal to a minus 20% of a which is equal to a minus 20 upon 100 into a Now from both these terms let us take a common so it will be a into 1 minus 20 upon 100 the whole which is equal to a into 1 minus 0.2 the whole that is equal to a into 0.8 the whole So amount after 1 year is equal to a into 0.8 the whole thus amount after t hours will be equal to a into 0.8 raised to power t Now here you can see that b is equal to 0.8 which is less than 1 it shows decay So general equation of the exponential growth is y is equal to a into 1 plus r upon 100 whole raise to power t where r is the person's rate of change or we can say growth rate Similarly general equation of exponential decay is y is equal to a into 1 minus r upon 100 whole raise to power t r is person's rate of change and say decay rate Now let us identify person's rate of change in the function given by y is equal to 1.01 whole raise to power t Now it can be written as y is equal to 1 plus 0.01 whole raise to power t which is equal to 1 plus 1 upon 100 whole raise to power t thus is equal to 1 percent increase in quantity it is 1 percent So in this session we have discussed how to use properties of exponents to interpret expression for exponential function and this completes our session hope you all have enjoyed the session