Answer to Question #194513 in Algebra for Rebira

Question #194513

Xlog x=100x


1
Expert's answer
2021-05-19T16:35:48-0400

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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