Logging Now 4

Algebra Level 4

$\frac{(\ln x)^2-3\ln x+3}{\ln x-1}<1$

If $x$ satisfy the inequality above, then find the domain of $x$ .

For more questions on logarithms try this set Logs of logs .

(1,2e)

(0,e)

No solution

(0,3e)

(1,-e)

1 solution

Yash Choudhary
Apr 5, 2015

$\Rightarrow \quad \frac { (lnx)^{ 2 }-3lnx+3 }{ lnx-1 } <1\\ Let\quad lnx\quad be\quad t,\\ \Rightarrow \quad \frac { { t }^{ 2 }-3t+3 }{ t-1 } <1\\ \Rightarrow \quad \frac { { t }^{ 2 }-4t+4 }{ t-1 } <0\\ \Rightarrow \quad \frac { { (t-2) }^{ 2 } }{ t-1 } <0\\ \therefore \quad t\in (-\infty ,1)\\ \Rightarrow \quad lnx<1\\ \Rightarrow \quad x>0\quad and\quad x<e\\ \therefore \quad \boxed { x\in (0,e) } .$

Why are questions in my set rated level 5 and all are above 245 pts.They are very easy.I thought that they would be level 2 or3. But i know there level decrease after some time. Nice solution.

Gautam Sharma - 6 years, 2 months ago

See now its level 2. :P

Yash Choudhary - 6 years, 2 months ago

If u r looking for something tough try this

Tanishq Varshney - 6 years, 2 months ago

@Tanishq Varshney – Nice problem.

Yash Choudhary - 6 years, 2 months ago

Same solution :D

Paul Ryan Longhas - 6 years, 2 months ago

over rated problem

Tanishq Varshney - 6 years, 2 months ago

Logging Now 4

For more questions on logarithms try this set Logs of logs .

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