Janet Forbes

2022-07-08

What is the number of real roots of $\left(\mathrm{log}x{\right)}^{2}-⌊\mathrm{log}x⌋-2=0$ $⌊\phantom{\rule{thinmathspace}{0ex}}\cdot \phantom{\rule{thinmathspace}{0ex}}⌋$represents the greatest integer function less than or equal to x.
I know how to solve logarithm equation but due to greatest integer function I am unable to proceed further please help thanks.

Since $\left[\mathrm{log}x\right]\le \mathrm{log}x$
we have $\left(\mathrm{log}x{\right)}^{2}-\mathrm{log}x-2\le 0$
This is equivalent to $-1\le \mathrm{log}x\le 2$
When $-1\le \mathrm{log}x\le 0,\left[\mathrm{log}x\right]=-1$ so that $\mathrm{log}x=±1$ If we see that $\mathrm{log}x=1$ is not in the specified range. Hence $\mathrm{log}x=-1$ and $x=\frac{1}{10}$
When $0\le \mathrm{log}x<1$, $\left[\mathrm{log}x\right]=0$ so that $\mathrm{log}x=±\sqrt{2}$ None of these values in the range.
Similarly we can use $1\le \mathrm{log}x<2$ this will give us $x={10}^{\sqrt{3}}$
When $\mathrm{log}x=2$, $\left[\mathrm{log}x\right]=2$ and equation is satisfied. Thus $x=100$ is third real root.

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