00:01
Hi, our first we understand here, we have this is y, this is x here, right? we just draw here the log graph, the graph of log x.
00:10
The domain is coming out to be here we have x greater than 0.
00:14
That is the domain we have, right? so now, what is given here in the a part, portion 1, in the a part, caution 1.
00:25
We want to find a domain of here, it is given that log of 4x square minus x2 power 4, 4x square minus extra power 4 with paces 4 right so we have the domain we'll have here 4 x square minus x 2 power 4 it is greater than 0 right so for that we get x square and then 4 minus x square is going to have to be greater than 0 or we get x square times will be 2 plus x 2 minus x is greater than 0 you can use now sign diagram here we equate each factor to 0 we get x equal to 0 x equal to minus 2 from here from here we get x equal to 2 right so we get now the next part is if we take a number to the right of 2 take here let's say x is equal to 3 this will become 9 then 2 plus 3 and 2 minus 3 that's coming out to be negative right we take a number here between 0 and 2 to take it as x equal to 1 let's suppose it will become 1 times 2 plus 1 and then 2 minus 1.
01:33
This is going to have to be positive.
01:36
Similarly, you take a number here between 0 and 2.
01:38
You take it as minus 1.
01:39
It's coming out to be positive and this will be negative here.
01:43
Now it is greater than 0.
01:44
You get here, the solution that is coming out to be from minus 2 to 2.
01:48
But the given problem is that is log 4xx minus x2 power 4.
01:54
Log 4x2 power 4, right? with base 4.
01:58
If it is equal to 0, then this will become log 0, which is not defined.
02:02
Right? we can't take x -go to 0 so therefore the interval is coming at to be from minus 2 to 0 union 2 to infinity i'm sorry minus 2 to 0 to 0 union we have 0 to 2 that is coming up to be the required domain for that minus 2 to 0 and then we have it's given as 0 to 2 that is the required domain let's move on further now to second one in second one would of find out the domain for log of x over 3 minus x the base 3 log of x over 3 minus x the base 3 right this we have here so again we can take here for the domain x over 3 minus x it is greater than 0 right this we have here so we just draw a sign diagram here right just draw that here put each factor to 0 so put x equal to 0 we get 0 here and then put x minus 3 equal to 0 you get x equal to 3 right so we have here and the next part is if we take a number here let's say 2 making the signs here take a number to the right of 3 but take x equal to 4 here it will become 4 over 3 minus 4 that's going to be negative here and similarly this will be positive this will be negative right for example if we take a number here to the left of 0 it's going to be negative if it take a number between zero and three it's come out to be positive right it's greater than zero here we have so solution that is given as we have zero to three that should be the solution set here right now think is that basically so zero and three both are excluded here right so both are excluded here so we can see solutions that is coming out to be zero to three so that is the required domain now i want to b part now here b part is we need to expand here that is given as log with base 2 or b part log with base 2.
04:20
Here we have x square minus x minus 2 all divided by this square root x square plus 3x minus 4 square we have x square plus 3x minus 4 so what we have here now we just solve it out we just expand the using the properties here.
04:54
So first what you can do here before expansion we can just factorize it.
04:58
So it's going to be log with base 2.
05:01
You can factorize x squared as x minus 2, that is given as x minus 2 and then x plus 1...