Question 1107750: Determine all the values of x for which:
√x^(log10x)=100
(To be read as "the square root of x to the power of log to the base 10 of x, equals 100")
I tried to solve this, but I keep getting different answers. I think x=100, but how do I find the other possible values of x?
Thank you in advance.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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.
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