Math Olympiad | A Nice Algebra Problem| Can You Solve this?☝️

log of x to the base 2 plus log of x to the base 5 is equal to 6. So we have to find out the value of x here. How we can find out the value of x? Let's see solution. log of x to the base 2 plus log of x to the base 5 is equal to 6. Right, this is question. So here we can, we can apply here one formula. If log of a to the base b is given, so we can write this one log of a divided by log of b. Right, so here log of x to the base 2. So we can write this one log of x divided by log of 2 and here log of x divided by log of 5. So we can write this one log of x divided by log of 2 plus log of x divided by log of 5 is equal to 6. Now log of x, log of x here. So we can take common log of x. So it will become here 1 upon log of 2 plus 1 upon log of 5 is equal to 6. Right, now we can write this one log of x divided times log of 2. We can do here cross multiplication. So log of 5 plus log of 2. Here log of 2 times log of 5. So log of 2 times log of 5 is equal to 6. Right, now log of 2 times log of 5. So we can multiply this term. So here we can write this one log of x times log of 5 plus log of 2, log of 2 divided by log of 2 times log of 5. This one as it is log of 5 is equal to 6. Now here log of a, log of a plus log of b is equal to, we can write this one log of a times b. Right, so we can write this one log of x times log of 5 times 2. Log of 5 times that is log of 10 divided by log of 2 times log of 5 is equal to 6. Now log of 10 to the base 10. If log of 10 to the base 10, so it will become here 1. Right, so log of x times 1 upon 1 upon log of 2 times log of 5 is equal to 6. You can do here cross multiplication. So log of x times log of x, sorry log of x divided by log of 5, it will become here log of 5 is equal to 6 divided, 6 times this one, 6 times log of 2 cross multiplication, 6 times log of 2. Right, now log of x divided by log of 5. So we can write this one log of x to the base 5, because we already seen this formula that is log of x divided by log of 5 it is given. So we can write this one log of x to the base 5. So here same way is equal to 6 times log of 2. Now we have to take here base 5 on both sides. We are taking here, so if we are taking base 5 both sides, so we will get here 5 power log of x to the base 5. We are taking base, so here again 5 power 6 times log of 2. Right, now again we have to use one formula that is if a power log of b to the base a is given, so where will be b. So here same thing when we are doing this formula we will get here value of b that is x. So it will become here x is equal to 5 power 6 log of 2, 6 log of 2. So this is the value of x here in this equation. Now let's verify this correct or not. So let's verify our value. Question is log of x to the base 2 plus log of x to the base 5 is equal to 6. This is question. Now we have to substitute value of x is 5 power 6 log of 2. Right, so here log of x, x is 5 power 6 log of 2 to the base 2. This one plus log of x again, so 5 power 6 log of 2 to the base 5 here. Question mark 6. Right, now log of 5 to the base 5, so it will become here 1. So we can write this one 6 log of 2 and one more formula we have to use here. If log of a power b is given, so we know this formula. This value will be b log of a. So using this formula here power is 6 log of 2. So it comes here, so 6 log of 2, 6 log of 2 times log of 5 to the base 2, log of 5 to the base 2 plus here value will be 1. So here it will become 6 log of 2. So 6 times log of 2, 6 log of 2 and here value will be 1. Question mark 6. Right, now 6 log of 2 here and 6 log of 2 here. So we can take common this. So 6 log of 2, we are taking common. So left term that is times of log of 5 power 2 plus 1, plus 1 this one, plus 1. Question mark 6. Now this is 6 log of 2, so we can write as it is 6 log of 2 times log of 5 to the base 2. So this log of 5 to the base 2, we can write log of a to the base b is to log of a divided by log of b. So using that formula we can write this one log of 5 divided by log of 2 plus 1 this one. Question mark 6. Now we can write this one 6 times log of 2, 6 times log of 2 times log of 5, this one log of 5 divided by log of 2 plus again log of 2, this one log of 2. Question mark 6. Now 6 log of 2 times log of 5 and here log of 2. So we can cancel this one, log of 2 cancel by log of 2 here. So left term that is 6 times log of 5 plus log of 2, this one. Question mark 6. Now we can write this one using same formula that is if log of a plus log of b is given. So we can write log of a times b. So this is the form of log of a plus log of b. So we can write this one log of 5 times 2. So we can write this one 6 times log of 5 times 2. Question mark 6. Now 6 times it will become here log of 10. Question mark 6. Now log of 10 to the base 10 value will be same that is 1. So we can write here 6 times 1. Question mark 6. So here it will become 6 times 1 is 6 is equal to 6. So we can say here left hand side is equal to right hand side. So here we can say value of x is equal to 5 power 6 log of 2. This is correct solution for this equation. It's verified here. Thank you.