 Hello and welcome to the session. In this session we will discuss the following question which says, solve for x log of 3 equals x to the base 3 plus log of 8 minus x to the base 3 minus log of 9 plus 9 minus 8 to the base 3 is equal to 2 minus log 9 to the base 3. Before moving on to the solution, let's discuss some laws of logarithms to be used in the solution. First we have log of x1 x2 to the base a is equal to log x1 to the base a plus log x2 to the base a. Then next law is log of x1 upon x2 to the base a is equal to log x1 to the base a minus log x2 to the base a. Then next law is log of x1 to the power n to the base a is equal to n into log x1 to the base a. log that a is a positive number and this a is not equal to 1 then x1 and x2 are also positive numbers and this n is n real number that is n belongs to r. This is the key idea that we use for this question. Let's proceed with the solution now. We are given that log of x3 plus log of to the base 3 minus log of 9x minus 8 to the base 3 is equal to log 9 to the base further log of to the base 3 plus log of to the base 3 minus log of 9x minus 8 to the base 3 is equal to 2 minus log of 3 square to the base. Using this law that is log of x1 x2 to the base a is equal to log x1 to the base a plus log x2 to the base a. These two terms log of to the base 3 is equal to we would use this law that is log of x1 to the power n to the base a is equal to n into log x1 we would get 2 into log n for the left hand side that is these two terms. We will use the law log of x1 upon x2 to the base a is equal to log x1 to the base a minus log x2 to the base a. This gives us log of and this whole upon 9x minus to the base 3 is equal to log of a to the base a is equal to log of 1 so log of 3 to the base 3 would be 1 and so 2 into 1 is 2 so here we have 2 minus 2. This gives us log of the whole into 8 minus x and this whole upon 9x minus to the base 3 is equal to 0 and we here is log 1 so log of 1 into and this whole upon 9x minus 8 and this to the base 3 is equal to log 1 to the base 3. So this means that 3 plus x into and this whole upon 9x minus 8 is equal to 1. For further cross multiplying we get 3 plus x into 8 minus x is equal to 9x minus 8. By multiplying these two terms we get 3 into 8 is 24 minus 3x plus 8x minus x square is equal to 9x minus 8 minus x square plus 5x plus 24 minus 9x minus x square plus 32 is equal to 0. Now splitting the middle term of the quadratic equation we get x square minus 4x is 32 is equal to 0. We have x into x minus 4 the whole plus 8 into x minus 4 the whole is equal to 0 which is equal to that either x is equal to 4 or x is equal to minus 8. These two solutions are correct or not. So for we consider that is log of to the base 3 plus log of to the base 3 log of 9x minus 8 to the base 3 is equal to log 7 to the base 4 to the base 3. We get 28 is equal to 2 to the power 2 into log 28 is equal to 7. Using the laws of logarithms stated in the key idea and further again using the law we get 20 log 2 plus log 7 to the base 3 plus log 4 to the base 3 log 7 to the base. Now this is equal to log 7 to the base 3 plus log 4 to the base 3 minus that this term can be written as log of 2 square minus log 4 to the base 3 minus log 7 to the base 3. Now these terms cancel and so we get this is equal to 0 that is the other choice is equal to 0. Now we consider the RHS 2 minus log 9 to the base 3. This can be written as 2 minus log 3 square to the base 3 3 to the base 3 plus log of A to the base A is equal to 1. So 2 into 1 is 0. That is we have the RHS is also equal to 0. Thus is equal to the RHS we say we consider the other solution which is it's equal to minus we get log minus 5 to the base 3 plus log 16 to the base 3 log of minus 18 to the base 3. Therefore is equal to minus.