Rex Maxwell

2022-04-03

Solving ${x}^{\mathrm{log}\left(x\right)}=\frac{{x}^{3}}{100}$
How do I find the solution to:
${x}^{\mathrm{log}\left(x\right)}=\frac{{x}^{3}}{100}$
So I multiplied 100 both sides getting:
$100{x}^{\mathrm{log}\left(x\right)}={x}^{3}$
Now what should I do?

kanonickiuoeh

I suppose log means ${\mathrm{log}}_{10}$? I'm not familiar with this sort of notation. Take logarithm on both sides, and you will get $2+{\mathrm{log}}^{2}x=3\mathrm{log}x$ Substitute $\mathrm{log}x$ with t. And you get ${t}^{2}-3t+2=0$, therefore $\left(t-1\right)\left(t-2\right)=0$. That should do it.

Marcos Boyer

Since ${\left(\mathrm{log}\left(x\right)\right)}^{2}=\mathrm{log}\left({x}^{\mathrm{log}x}\right)=\mathrm{log}\left(\frac{{x}^{3}}{100}\right)=3\mathrm{log}\left(x\right)-2$, we have ${\left(\mathrm{log}\left(x\right)\right)}^{2}-3\mathrm{log}\left(x\right)+2=0$. Hence, $\mathrm{log}\left(x\right)=2$ and $\mathrm{log}\left(x\right)=1.$
Therefore, $x=100$ atau $x=10$

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