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Question:

Solve for $x:\log_x(\frac52-\frac1x)\gt\frac52-\frac1x$

My approach:

I tried taking two cases when $x\gt1$ or when $0\lt x\lt1$. For the first case, I wrote $\frac52-\frac1x>x^{\frac52-\frac1x}$. But for this case, the book wrote $\frac52-\frac1x\gt x$. Why? Also, $\frac52-\frac1x\gt0\implies\frac52\gt\frac1x\implies x\gt\frac25$

I tried making graph but couldn't conclude anything.

Soumil Gupta
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aarbee
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1 Answers1

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I drew a graph and it looks like this: Desmos graph

So as you can see, $0.4<x<0.546$ and $1<x<1.776$ seem to be the solutions. I am not sure if this can be done in an analytical way because I feel that there is SOME sort of $x^x$-like expression involved.

As for why the book wrote that, I feel that is because they are not trying to find a very accurate bound. For $x>1$, $\frac{1}{x}<1 \Rightarrow \frac{5}{2}-\frac{1}{x}>1$, meaning that $x^{\frac{5}{2}-\frac{1}{x}} > x$.

Prometheus
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