Math Olympiad| Nice exponential math problem| solve for x

Let's quickly solve for the value of x. This is a very tricky one but we'll get it together. So 3 to the power of x plus 3 to the power of x, they have 3 to the power of x in common right? Now when this is divided by this we have 1 plus when this is divided by this we have another 1 is equal to 19. Okay so this simply means 2 multiplied by 3 to the power of x is equal to 19. Now let's go ahead and divide both sides of the equation. So when we do that divided by 2 divided by 2 we have that 3 to the power of x is equal to 4 to 5. Now I would like you to stick to the end of this video because the way I proved that what we got is right using logarithm is very amazing. You will be glad you watched the end. So I advise you watch the end and see how I put the value back into this equation and got it correctly. Okay all right and got the 90 to show that what we got is right. Now at this point we all know that we just have to introduce log. Okay so let's take logarithm of both sides of this equation. So if we do that we have log 3 to the power of x is equal to log 45. Okay now when you have log a to the power of b is same thing as b multiplied by log a. Okay so using the power rule we are going to bring our x down. So we'll have x log 3 is equal to log and 45 is same thing as 9 times what 5 log 9 times 5. All right perfect. Now let's divide both sides of this equation by log 3. So we have x log 3 divided by log 3. Okay is equal to log 9 multiplied by 5 divided by log 3. So this we divide this and we have that x is equal to log. Now 9 multiplied by 5 when you have log a b is same thing as log a plus log b. All right so we are going to express this in this form. So let's go ahead and do that. So what we have is log 9 plus log 5 divided by log 3. Okay so let's continue we have that x is equal to log 9 is same thing as 3 squared right plus log 5 divided by log 3. So we have that x is equal to this is same thing as log 3 squared divided by log 3 right plus log 5 divided by log 3. Perfect. If this is the first time you are coming across this channel please click on the subscription button and on your notification bell so that you'll be notified anytime I upload a new video. All right I upload two videos each day so you can come at any time to check. I must upload twice by the grace of God daily. All right now let's continue. So at this point we have that x is equal to let's bring this down as usual so we have 2 log 3 over log 3 right plus log 5 over log 3. Now log 3 we divide log 3 and we have that x is equal to 2 plus log now log 5 log 3 same thing as log 5 base 3. Now this is our x like I said the amazing part of this thing is where we have actually proved this that this is the solution. Now let's prove that okay so the equation says that 3 to the power of x plus 3 to the power of x is equal to 90. So let me clean this. Let's clear everywhere and prove that what we got is actually right. Okay so we have 3 to the power of x plus 3 to the power of x is equal to 90. Now recall that our x is 2 log 5 base 3 right so we have 3 raised to the power of 2 plus log 5 base 3 plus 3 raised to the power of 2 plus log 5 base 3 is equal to 90 is equal to 90. All right so we have this is same as using our law of indices right this same thing as 3 to the power of 2 multiplied by 3 to the power of log of base 3 log 5 base 3 plus we are going to do the same thing here 3 squared multiplied by 3 raised to the power of log 5 base 3 is equal to 90. Perfect so what we have here is 9 multiplied by so when you have a raised to the power of log a b all right this is equal to b when you have something like this this is equal to b and you have something similar here that means the whole of this is equal to 5. Okay so we are going to write 9 multiplied by 5 plus the same thing here 9 multiplied by 5 is equal to 90. Now what is 9 multiplied by 5 that is 45 plus 45 is equal to 90 and here we have that 90 is equal to 90. This proves that our solution is right. Isn't that amazing? Thank you so much for watching and do have a lovely day.