Solving a 'Harvard' University Entrance Exam Question

Hello everyone you are welcome. How to solve and find the value of x in this interesting logarithm math problem which is log of x with base 2 plus log of x with base 5 is equal to 6. Now how can we solve this log from a different basis. Now first of all here in the left hand side we will use change of base logarithm identity. So here in the left hand side we will use this one we can write log of a with base b as log of a divided by log of b. So using this logarithm property here this left hand side will become this implies this will become log of x divided by log of 2 plus this will become log of x divided by log of 5 is equal to 6. If there is log of x common in the numerator so we will take out common this will become taking log of x common so this is 1 by log of 2 plus and this will become 1 by log of 5 is equal to 6. Here we will add these two numbers so this will become log of x here we will see log of 2 times log of 5. Dividing this number by log of 2 gives log of 5 we will multiply it with 1 so this will become log of 5 plus log of 2 is equal to 6. Here inside the numerator we will use another logarithm property. So here in the numerator we will use this one logarithm property we can write log of a times b as log of a plus log of b. So using this logarithm property here this numerator will become this implies log of x times and this will become log of 5 times 2 which is log of 10 divided by log of 2 times log of 5 is equal to 6. Here this is log of 10 with base 10 and as we know that log of any number with the same base is always equal to 1 so here we will replace this number with 1 so we multiply this log of x with 1 this will become log of x and here we will move this log of 2 to the right hand side so it will be multiplied with 6. So this equation will become this will become log of x divided by log of 5 is equal to 6 times log of 2. Here in the left hand side in this number again we will use this one logarithm property and we will change that number from this one form to this one form. So here this equation will become simplifies this will become log of x with base 5 is equal to 6 times log of 2. Here we will take 5 in base in both sides so this equation will become this will become 5 raised to power log of x with base 5 is equal to 5 raised to power 6 times log of 2. Here in the left hand side we will use another logarithm identity another logarithm property so here in the left hand side you will use this one logarithm property we can write a raised to power log of b with base a is equal to b so using this logarithm property here we will replace this left hand side with x so this left hand side will become x is equal to 5 raised to power 6 times log of 2. So finally x is equal to 5 raised to power 6 times log of 2 is our answer and value of x in this interesting logarithm problem and we will try to verify this value of x data x is equal to 5 raised to power 6 times log of 2 is the exact and correct value of x in this problem or not so we will verify this value here so let's verify to verify the value of x here we will write our problem again our problem is log of x with base 2 plus log of x with base 5 is equal to 6 so here we will replace x with this value so this left hand side will become this will become log of here the value of x is 5 raised to power 6 times log of 2 with base 2 plus this will become log of base 5 with x is 5 raised to power 6 times log of 2 is equal to 6 here inside these two numbers we will use a logarithm math property so here at this stage we will use this one logarithm math property here we can write log of a raised to power n as n times log of a so using this logarithm property here we will move these two powers to the front of log so this left hand side will become this will become 6 times log of 2 log of 5 with base 2 plus and this will become 6 times log of 2 times this will remain log of 5 with base 5 and we know that log of any number with the same base is 1 so this will become 1 is equal to 6 notice 6 times log of 2 common in both the terms in the left hand side so we will take out common so taking log 6 times log of 2 common this will become log of 5 with base 2 plus this will become only 1 is equal to 6 here inside this one number we will use this one logarithm property so therefore we can write this number as this is 6 times log of 2 times and this number will become log of 5 divided by log of 2 plus 1 is equal to 6 here we multiply this log of 2 inside the parentheses with both the terms this will become 6 times this will become log of 2 times log of 5 divided by log of 2 plus log of 2 is equal to 6 we have this log of 2 and this log of 2 will be cancelled this will become 6 times log of 5 plus log of 2 is equal to 6 we have this expression inside the parentheses as in this one form so therefore we can write this as log of the product of these two numbers so this will become log of 5 times 10 this is 6 times this will become log of 5 times 2 is 10 so this will become log of 10 is equal to 6 this is 6 times and this is here the base is 10 so this is equal to 1 is equal to 6 the 6 times 1 is simply 6 is equal to 6 since both sides are equal so it means that x is equal to 5 raised to power 6 times log of 2 is the exact and correct value of x in this nice and interesting logarithm math problem now in last don't forget to subscribe my youtube channel for such a more interesting and step-by-step math videos thanks for watching