The fastest way to solve this exponential equation.

To solve this we actually need to understand the laws of exponents. Welcome back to my channel. So this looks complex but it's actually very very easy. Though it's tricky. The first thing we need to do is to make this a linear equation by multiplying both sides by 4. So if we multiply both sides by 4 we have 4 multiplied by 2 to the power of x multiplied by 2 to the power of x divided by 4 is equal to 10 multiplied by 4. Now this 4 is going to take away this 4 right and we are left with 2 to the power of x multiplied by 2 to the power of x is equal to 10 multiplied by 4. Now remember that 8 to the power of m multiplied by 8 to the power of m is actually 8 to the power of m all squared. Therefore we can write this as 2 to the power of x all squared is equal to 10 multiplied by 4. Now if this is the first time if I forget if this is the first time you have seen this wonderful channel please click the subscription button turn on the notification bell so that you don't miss us okay. If you don't this might be the last time you're going to see this channel and this face and I wouldn't like to miss it all right. So if you have anything to tell us about the video please we appreciate it very well drop your comments. I'm always in the comment section and always turning out your comments you must continue to watch by the grace of God. Thank you so much all right. Now looking at this we have 2 to the power of x raised to the power of 2 is equal to 10 multiplied by 4 but 8 to the power of m raised to the power of n is equal to 8 to the power of m times n okay. Now we are going to have a 2 to the power of x raised to the power of 2 right. Oh what am I doing So we're going to have I'm repeating this which is unnecessary. So we're going to have 2 to the power of x times 2 is 2x is equal to 10 multiplied by 4. Yeah shout out to all Catholics all over the world as we celebrate the nativity of the Blessed Virgin Mary and I pray that all the graces that we need here on earth should be granted to us through our intercession. Amen okay let's continue. So now we have 2 to the power of 2x is equal to 10 multiplied by 4. Now let's introduce log to both sides of the equation because that is the only option we have since the value since the x that we are looking for is our power. So introducing log on both sides we have log 2 to the power of 2x is equal to log 10 multiplied by 4. Okay so now remember that log a raised to of m is equal to m multiplied by log a. So it simply means that we are going to write this as 2x multiplied by log 2 is equal to log 10 multiplied by 4. Now let's divide both sides by log 2 okay divided by log 2 divided by log 2. Now this is going to take away this and we are left with 2x is equal to log 10 multiplied by 4 divided by log 2. Now I want you to recall that log a multiplied by b is equal to log a plus log b. Now what that means is that we can write this as 2x is equal to log 10 plus log 4 everything divided by log 2. Alright have you subscribed? If you have not then I think you need to do it now so that you don't forget right. Now recall that when you say that a plus b when you have a plus b divided by c that it is equal to a divided by c plus b divided by c. So that means that we can actually write this as 2x is equal to log 10 divided by log 2 plus log 4 divided by log 2. Okay now if we proceed if we proceed we will have remember that 10 we can write 10 as 2 multiplied by 5 and we can write 4 as 2 multiplied by 2 which is also 2 squared. So that means that we can write this as 2x oops so we can write that as 2x is equal to log 2 multiplied by 5 divided by log 2. Okay then plus this will give us log 2 squared divided by log 2. See this place was wet. Okay now here we are going to apply the law that log ab is equal to log a plus log b and we are going to write log remember that also log a raised to the power of b is equal to b multiplied by log a. These two laws we are going to use it to expand this. We are going to prove that the answer we got is correct. Okay so our advice I would ask that you stay till the end all right so that you will enjoy the video all right because that is where the video gets more they understand. So now for here we are going to write that 2x is equal to log 2. So we have log 2 plus log 5 right log 2 plus log 5 using this law divided by log 2 then plus now here these two we come down according to the second law right so we have 2 log 2 divided by log 2. So if we distribute log 2 here we have 2x is equal to log 2 divided by log 2 plus log 5 divided by log 2 plus now until we go into itself one then we have only two this side and here log 2 we divide itself we have one so what we actually have here is 2x is equal to 1 plus log 5 divided by log 2 plus 2. Okay now if we simplify further you see how beautiful the answer is coming out so if we simplify this further we have 1 plus 2 which is 3 right and if we apply change of base remember that log a over log b is equal to log a is b. So we are going to have 1 plus 2 is 3 so we have 2x is equal to 3 plus this becomes log 5 base 2. Now we need x but not 2x therefore we need to divide both sides by 2 divided by 2 divided by 2 this we go into itself once and we are left with x is equal to 3 plus log 5 base 2 log 5 base 2 divided by 2. So this becomes our answer. I don't know if you can see that clearly but I think you can. Now we are going to prove that the answer we got is correct but like I said if you watch to this stage congratulations because you are about to enjoy the whole show more. I know you've been enjoying it but you're going to enjoy it more now. Yes let us hit this hit the nail on the head so instead of us to write this in this form remember that we simplify this as 2 to the power of x all squared right divided by 4 is equal to 10. So in this form now it is easier for me. Now I'm going to substitute my x so in order to make that easier I'm going to switch this to 2 and this to x. Yes because a to the power of m raised to the power of n is also equal to a to the power of n raised to the power of m because you just have to multiply this to remove the bracket. So m times n and n times m still the same thing. So now we'll write 2 to the power of 2 brackets our x now is this 3 plus log 5 base 2 everything divided by 2. All right now divided by 4 is equal to 10. If you look at this you will notice that 2 is here and 2 is there and when this multiplies what we have in the bracket this 2 goes out. So we are now so we are now left with 2 to the power of 3 plus log 5 base 2 everything divided by 4. Now is this equal to 10? Recall that a to the power of m plus n is equal to a to the power of m multiplied by a to the power of n. So if we substitute this into this or if we write this in this form sorry we'll write 2 to the power of 3 multiplied by 2 to the power of log 5 base 2 divided by 4. Now is this equal to 10? 2 to the power of 3 is 8. So we have 8 multiplied by now recall that a to the power of log b base a is equal to b. So once they share same base it is b. So and that is what we have here that means that the whole of this is 5 divided by 4 and this is this equal to 10. So for we go into it two times right and 2 times 5 is 10. So you can see that 10 is equal to 10 and that proves that the answer you got is correct. Thank you so much for watching and see you in my next video. Bye!