10^x=x^10 find x?
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10^x = x^10
There is only one solution for the function which is:
x = 10
10^10 = 10^10
To find the solution we will graph both functions and see where they intersect.
The point in which both functions intersect is the solution for the probelm.
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10^x = x^10.
Solution:
The function has an obvious solution at x= 10 . But It has other solution than x =10. Let us examine.
Let f(x) = 10^x - x^10.
f(1) = 10^1 -1^10 = 9 > 0
f(2) = 10^2 -2^10 = 100 -1024 = -924.
Therefore f(x) being a continuous and derivable function f(x) = 0 has a solution in the interval (1 , 2) as it changes the sign for x=1 and x =2.
We can go for an iteration method and have a solution for x.
When x = 1.371289,
10^x =23.51197 and x^10 =23.51202. So f(x) < 0
when x= 1.371288
10^x = 23.51191 and x^10 = 23.51185 . Here f(x) > 0.
So clearly there is a solution for x in between 1.371288 and 1.371289 where f(x) change the sign.
The equation will have only one solution, namely
x = 10
10^10 = 10^10
To prove the uniqueness of this solution, the equation could be solved drawing the graphs of the exponential function, 10^x, and the polynomial function, x^10.
The intercepting point of these 2 graphs is the solution of the equation.
To draw each graph, we'll have to input values to x.
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