Xlog x=100x
As the usage of "\\log x" requires "x\\neq 0", we can divide both sides by "x" knowing that it is non-zero.
"x^{\\log x}\/x = 100"
As "a^b\/a^c = a^{b-c}" and "a=a^1", we have
"x^{\\log x}\/x = x^{\\log(x)-1}"
"x^{\\log (x)-1}=100"
As we can write "a^b=10^{b\\log a}", we have
"10^{(\\log(x)-1)\\cdot\\log(x)}=100"
Finally, as "100=10^2" we conclude
"(\\log(x)-1)\\cdot\\log(x) = 2"
"\\log^2(x)-\\log(x)-2=0"
Solving this equation quadratic in "\\log(x)" gives us
"\\begin{cases}\n\\log(x_1)=2 \\\\\n\\log(x_2)=-1\n\\end{cases}"
And by removing the log we obtain two solutions :
"\\begin{cases}\nx_1=100 \\\\\nx_2=0.1\n\\end{cases}"
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