Question 1107750
your equation is sqrt(x ^ log(x)) = 100


base 10 is implied if no base is shown.

swuare both sides of this equation to get:


x ^ log(x) = 10,000.


take the log of both sides of this equation to get:


log(x ^ log(x)) = log(10,000).


log(x ^ log(x) is equal to log(x) * log(x) which is equal to log(x)^2.


you get log(x)^2 = log(10,000).


log(10,000) is equal to 4.


therefore log(x)^2 = 4.


take the square root of both sides of this equation to get log(x) = plus or minus 2.


log(x) = 2 if and only if 10^2 = x.


that makes x = 100.


log(x) = -2 if and only if 10^-2 = x.


10^-2 is equal to 1/100.


therefore 1/100 = x.


looks like x can either be 100 or 1/100.


replace x with each of these values to see if the original equation holds true.


the original equation is sqrt(x ^ log(x)) = 100


when x = 100, this becomes sqrt(100 ^ log(100)) = 100.


since log(100) = 2, this equation becomes sqrt(100 ^ 2) = 100.


since sqrt(100^2) = 100, you get 100 = 100, confirming that x = 100 is a solution to the original equation.


now let x = 1/100.


the original equation is sqrt(x ^ log(x)) = 100


when x = 1/100, the original equation becomes sqrt((1/100) ^ log(1/100)) = 100.


log(1/100) = -2, therefore you get sqrt((1/100) ^ -2) = 100.


(1/100) ^ -2 is equal to 10,000.



(1/100)^ -2 = 10,000.


the equation becomes sqrt(10,000) = 100.


this results in 100 = 100 which is true, confirming that x can either be 100 or 1/100.


that's your solution.