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May 1st, 2014, 01:26 AM  #1 
Member Joined: Sep 2013 Posts: 75 Thanks: 0  logarithm inequality
Log base 7(3x  1) + log base 7 (2x + 1) < 0. I understand that (3x  1) >0 and 2x + 1> 0. That gives me x > 1/3 and x >  1/2 but the second inequality is wrong how?

May 1st, 2014, 02:05 AM  #2 
Senior Member Joined: Apr 2014 From: Europa Posts: 584 Thanks: 177 
$\displaystyle \color{blue}{x>\dfrac{1}{3}\Longrightarrow\ x \in (\dfrac{1}{3},\ \infty )\ \qquad (1) \\\;\\ AND \\\;\\x>\dfrac{1}{2}\Longrightarrow\ x \in (\dfrac{1}{2},\ \infty)\ (2)\\\;\\(1), \ (2)\ \Longrightarrow\ x \in (\dfrac{1}{3},\ \infty ) \cap (\dfrac{1}{2},\ \infty) \Longrightarrow\ x \in (\dfrac{1}{3},\ \infty )}$
Last edited by aurel5; May 1st, 2014 at 02:09 AM. 
May 1st, 2014, 05:48 AM  #3 
Member Joined: Sep 2013 Posts: 75 Thanks: 0 
The second inequality is not correct according to my answers

May 1st, 2014, 05:57 AM  #4 
Senior Member Joined: Apr 2014 From: Europa Posts: 584 Thanks: 177  
May 1st, 2014, 05:58 AM  #5 
Member Joined: Sep 2013 Posts: 75 Thanks: 0 
The second inequality is not correct according to my answers. I got the same answer as yours. Does that mean we are wrong or my book is wrong. The answer is 1/2<x >1/3 bt how???

May 1st, 2014, 06:22 AM  #6 
Senior Member Joined: Apr 2014 From: Europa Posts: 584 Thanks: 177 
$\displaystyle (\dfrac{1}{3},\ \infty)$ it is the field of existence (of inequality). Inequality solution (after some calculations ! ) is the interval $\displaystyle (\dfrac{1}{3},\ \dfrac{1}{2})\\\;\\ So,\qquad \dfrac{1}{3} <x < \dfrac{1}{2}$ 
May 1st, 2014, 06:56 AM  #7 
Senior Member Joined: Apr 2014 From: Europa Posts: 584 Thanks: 177 
$\displaystyle \color{blue}{Conditions\ for\ logarithms :\\\;\\3x1 > 0 \Rightarrow x>\dfrac{1}{3}\Longrightarrow\ x \in (\dfrac{1}{3},\ \infty )\ \qquad (1) \\\;\\ 2x+1 > 0 \Rightarrow x>\dfrac{1}{2}\Longrightarrow\ x \in (\dfrac{1}{2},\ \infty)\ (2)\\\;\\(1), \ (2)\ \Longrightarrow\ x \in (\dfrac{1}{3},\ \infty ) \cap (\dfrac{1}{2},\ \infty) \Longrightarrow\ x \in (\dfrac{1}{3},\ \infty )\ \ (3)\\\;\\ \log_7(3x1)+\log_7(2x+1) < 0\ \Rightarrow\ \log_7(3x1)(2x+1) < 0 \\\;\\(3x1)(2x+1) < 7^0 \Rightarrow 6x^2+3x2x1 < 1 \\\;\\6x^2+x2 < 0 \Rightarrow x \in (\dfrac{2}{3},\ \dfrac{1}{2})\ \ (4)\\\;\\(3), (4) \Rightarrow x \in (\dfrac{1}{3},\ \infty) \cap (\dfrac{2}{3},\ \dfrac{1}{2}) \Rightarrow x \in (\dfrac{1}{3},\ \dfrac{1}{2}) \Rightarrow \dfrac{1}{3} < x < \dfrac{1}{2}}$


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inequality, logarithm 
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