 Hello and welcome to the session. In this session we will discuss the change of base in log rhythms. Now we have to prove that log N to the base A is equal to log N to the base B into log B to the base A and here N, A and B are all positive numbers. Now let us start with its proof. Now let log N to the base A is equal to X and log N to the base B is equal to Y and log B to the base A is equal to Z. Now using the definition of logarithm that is if three numbers A, X and Y are written as A is super X is equal to Y and in logarithmic form we can write them as log Y to the base A is equal to X. So by using the definition of logarithm in these three equations for the first equation we will have A raised to power X is equal to N for the second equation we will have B raised to power Y is equal to N and for this equation we will have A raised to power Z is equal to B that is A raised to power X is equal to N and here N is equal to B raised to power Y and also B is equal to A raised to power Z. So this is equal to A raised to power Z whole raised to power Y which is equal to A raised to power YZ. This implies A raised to power X is equal to A raised to power YZ. Now here the bases are same so we can compare the parts. So this implies X is equal to YZ. Now using the values of X, Y and Z this equation will be log N to the base A is equal to log N to the base B into log B to the base A. So this formula changes the base from A to B. Now in the same way we can prove that log N to the base B is equal to log N to the base A into log A to the base B that is here we are also changing the base by using this formula also from this formula now let it be equation number one. So from one N to the base B is equal to log N to the base A over log B to the base A. Now let it be equation number two. So from two we have log A to the base B is equal to log N to the base B over log N to the base A and also log B to the base A is equal to log N to the base A over log N to the base B. Now this is the equation number one so for N is equal to A in the formula which is given by equation number one log A to the base A is equal to log A to the base B into log B to the base A. Now log A to the base A is equal to 1 so this implies 1 is equal to log A to the base B into log B to the base a. Now this is a very important result which is to be remembered. Now let us discuss some corollaries. Now the corollary 1 is log y to the base s is equal to log a to the base x over log a to the base y. That is by using this result which we have discussed earlier. Now by using this result which we have discussed earlier, we are getting the results that is log a to the base x is equal to 1 over log x to the base a whole upon log a to the base y is equal to 1 upon log y to the base a. So this is equal to log y to the base a whole upon log to the base a. Therefore log y to the base x is equal to log y over log x. So we can write this as log y over log x as both are having the same bases and also we have y, x and a are greater than 0 and also all these three are not equal to 1. Now let us discuss one example. Here log 18 to the base 2 can be written as. Now by using this result it can be written as log 18 to the base 10 log 2 to the base 10. Now log 18 to the base 10 is equal to 1.2553 over log 2 to the base 10 is equal to 0.3010 which is equal to 4.17043. This is how by using this result we can solve log 18 to the base 2. Now let us discuss the second problem is log a to the base b into log b to the base c into log c to the base a is equal to. Now by using this result which we have discussed earlier. Now log a to the base b can be written as log a to the base m over log b into. Now log b to the base c can be written as log b to the base m over log c to the base m into this can be written as log c to the base m over log 2 turning this will give 1. Therefore log a to the base b into log b into the base c into log c to the base a is equal to 1. Now let us discuss this problem which is log y raise to power m whole to the base x raise to power n is equal to log y raise to power m whole upon log x raise to power m to the base a. Now using the third law of logarithm log m raise to power n to the base a is equal to m log m to the base a. So this is equal to log y to the base a whole upon m log x to the base a. Now again using this result. So here log y to the base a over log x to the base a will become log y to the base x. So this is equal to m over n into log y to the base x. Now let us discuss one example. Now log 125 to the base 4 can be written as log 5 raise to power 3 to the base 2 raise to power 2. Now this is of the form log y raise to power m whole to the base x raise to power n. So here m is 3, m is 2, y is 5 and x is 2. So by applying this formula this will be equal to 3 by 2 into log y to the base x that is log 5 to the base 2. Now let us discuss the next problem which is x raise to power log y to the base a is equal to x raise to power log log y to the base x into log to the base a which is further equal to raise to power log y to the base x whole raise to power log x to the base a which is further equal to y raise to power log x to the base a. Now here we are using the change of base formula and here x raise to power log y to the base x is equal to y because x are positive real numbers and also a is not equal to 1 then a raise to power log x to the base a is equal to x. So in this case also x and y are positive real numbers and also x is not equal to 1 this implies x raise to power log y to the base x is equal to y. So these were the various corollies and in this session you have learnt about the change of base formula in logarithms. So this completes our session hope you all have enjoyed this session.