 Hello and welcome to the session. In this session we discussed the following question which says if log bc to the base a equal to x, log ca to the base b equal to y and log ab to the base c is equal to z, prove that 1 upon x plus 1 plus 1 upon y plus 1 plus 1 upon y plus 1 is equal to 1. Before we move on to the solution, let's discuss some results to be used in the solution. First we have a law of logarithm which is log x1, x2 to the base a is equal to log x1 to the base a plus log x2 to the base a. There x1 and x2 are positive numbers, a is also a positive number but a is not equal to 1. The second we have the change of base formula according to which log m to the base a is equal to log m to the base b into log b to the base a. This is the key idea that we use for this session. Let's now move on to the solution. We are given that log bc to the base a is equal to x, log ca to the base b is equal to y and log ab to the base c is equal to z. Now we are supposed to x plus 1 plus 1 upon y plus 1 plus 1 upon z plus 1 is equal to 1. For this we have the RHS that is 1 upon x plus 1 plus 1 upon y plus 1 plus 1 upon z plus 1 and we substitute the values for x, y and z. This is equal to 1 upon log bc to the base a that is x plus 1 plus 1 upon y which is log ca to the base b plus 1 1 upon z which is log ac plus 1 as this is equal to 1 upon log bc to the base a log of a to the base a with the terms that is both the log base b plus now 1 can be written as log b to the base b base b so we write 1 ac plus 1 can be written as log c to the base of this term so for this we can use this key idea which is log x1 x2 to the base a is equal to log x1 to the base a plus log x2 to the base the written as 1 upon log bc to the base of the second term we can write this as 1 upon log abc base b log abc to the base c formula we have log n to log n to the base b into log b to the base a log a to the base a is equal to log a to the base b into log b to the base a that 1 is equal to log a to the base b into log b to the base a since we know that log of a to the base a is equal to 1 and from this we get log a to the base b is equal to 1 upon log b to the base a let this be the result mhs abc plus 1 upon log abc to the base b can be written as log to the base abc plus log abc to the base c can be written as to the base abc so this is using the result now again as the lhs is the sum of the logs we will use this log logarithm that is log x1 x2 to the base a is equal to log x1 to the base a plus log x2 to the base from this we get this is equal to log abc to the base abc and this is equal to as we know that log of a to the base a is equal to 1 thus we have the lhs is equal to 1 which is the same we now have 1 upon x plus 1 plus 1 upon y plus 1 plus 1 upon y plus 1 is equal to 1 this is what we were doing.