Question

show the following equation has at least one solution
x-(lnx)^x = 0 [4,5]

Answer 1

it is not an equation yet, can you make sure that you did copy it right from your paper? An equation is of the form expression=expression, you have \(x-(\log x )^x\) is it supposed to be 0 ?

Answer 2

Good I can see you updated your equation now it is in the given form. So you try the the following set \(f(x)=x- ( \log x )^x\) and evaluate it at the given endpoints: \[\Large f(4)=4- (\log 4)^4\approx 0.30 >0 \\ \Large f(5)=5- ( \log 5)^5 \approx -5.79<0 \]
So you have found two values within your given set \(A=[4,5]\) in which the function takes on positive and negative values. Since \(f(x)\) is continuous on \(\mathbb{R}\) you can use the intermediate value theorem.

That means \(\exists x^*\in [4,5] : f(x^*)=0 \) which would introduce your solution

Answer 3

suppose f(x)=x-(lnx)^x
then
if f(4)*f(5)<0
will haveat least one solution

Answer 4

\( \Large f(x^*)=x^* - (\log (x^*))^{x^*}=0 \)