1. ## Log

$\displaystyle \log_2 3+\log_3 4+\log_4 5+\log_5 6+\log_6 7+\log_7 8$ How to find the value for this?

Find the range for x?

$\displaystyle 3x+4<x^2-6x<9-2x$

2. ## Re: Log

what have you done with either of these questions?

3. ## Re: Log

first one i can't done more but in second one i solve finally i get answers in root then i can't do more>>

4. ## Re: Log

Originally Posted by srirahulan
first one i can't done more but in second one i solve finally i get answers in root then i can't do more>>

5. ## Re: Log

First One,

$\displaystyle \log_2 3+\log_3 4+\log_4 5+\log_5 6+\log_6 7+\log_7 8$

$\displaystyle \log_2 3+\frac{2}{\log_2 3}+\frac{1}{\log_5 4}+\log_5 6+\frac{1}{\log_7 6}+\log_7 8$

And then What Can i Do?

Second One,
$\displaystyle 3x+4<x^2-6x<9-2x$

Case 1,
$\displaystyle 3x+4<x^2 -6$
$\displaystyle 0<x^2-3x-10$
$\displaystyle 0<(x-5)(x+2)$
In this case i get $\displaystyle x>5$Or$\displaystyle x<-2$

Case 2,
$\displaystyle x^2-6x<9-2x$
$\displaystyle x^2-4x-9<0$
$\displaystyle x=\frac{4 \pm \sqrt{16+36}}{2}$
then i get answers in roots so how can i find the range of x?

6. ## Re: Log

Originally Posted by srirahulan
first one i can't done more but in second one i solve finally i get answers in root then i can't do more>>
I see no simple way to do the first one.

It is equivalent to $\displaystyle \sum\limits_{k = 2}^7 {\frac{{\ln (k + 1)}}{{\ln (k)}}}$ .

Using a CAS I get 7.276.

7. ## Re: Log

I can't understand you solution or the first one..

8. ## Re: Log

Originally Posted by srirahulan
Second One,
$\displaystyle 3x+4<x^2-6x<9-2x$

Case 1,
$\displaystyle 3x+4<x^2 -6\color{red}x?$
$\displaystyle 0<x^2-3x-10$
$\displaystyle 0<(x-5)(x+2)$
In this case i get $\displaystyle x>5$Or$\displaystyle x<-2$

Case 2,
$\displaystyle x^2-6x<9-2x$
$\displaystyle x^2-4x-9<0$
$\displaystyle x=\frac{4 \pm \sqrt{16+36}}{2}$
then i get answers in roots so how can i find the range of x?
$\displaystyle 3x+4<x^2-6x<9-2x$

case (1) ...

$\displaystyle 3x+4 < x^2-6x$

$\displaystyle 0 < x^2-9x-4$

$\displaystyle x < \frac{9-\sqrt{97}}{2} \approx -0.4244...$

$\displaystyle x > \frac{9+\sqrt{97}}{2} \approx 9.4244...$

case (2) ...

$\displaystyle x^2-6x < 9-2x$

$\displaystyle x^2-4x-9 < 0$

$\displaystyle 2-\sqrt{13} < x < 2+\sqrt{13}$

$\displaystyle -1.60555... < x < 5.60555...$

intersect the two solution sets