November 8th, 2016, 11:36 AM  #1 
Newbie Joined: Nov 2016 From: Israel Posts: 1 Thanks: 0  A super hard logarithm problem
Hey, it's my first time visiting in a math forum but I seriously need help. Since there are no active forums in hebrew (I'm from israel) I went looking for english forums, like this one. So my math teacher gave it to us today saying he will bump up the grade of anyone solving it, showing they way of solving, obviously. Anyway, this is the problem: Anyone solving or at least trying to solve it, be blessed! thank you. *explain how you did it. the final result is 9, as he told us, but the way is what matters. 
November 8th, 2016, 02:20 PM  #2 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,659 Thanks: 652 Math Focus: Wibbly wobbly timeywimey stuff.  Quote: Dan 
November 8th, 2016, 04:58 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,640 Thanks: 959 Math Focus: Elementary mathematics and beyond 
After some manipulation and using some of the rules of logarithms one may arrive at $$\left(\frac23\right)^y=\frac{\sqrt x+1}{x}$$ which has an integer solution at $x=9$. 
November 8th, 2016, 05:16 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,145 Thanks: 1418 
Let x = 9^t, so that the equation becomes 4^t = 3^t + 1. By inspection, t = 1 is a solution, and so x = 9. Show that no other solution exists. 
November 9th, 2016, 09:17 AM  #5  
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  Quote:
$\displaystyle (2^t  1)(2^t + 1) \ = \ 3^t$ For possible integer solutions, $\displaystyle (2^t  1)(2^t + 1) \ = \ 3\cdot3^{t  1}$ $\displaystyle 2^t  1 \ = \ 3 $ $\displaystyle 2^t + 1 \ = \ 3^{t  1}$  or it's the case $\displaystyle 2^t  1 \ = \ 3^{t  1}$ $\displaystyle 2^t + 1 \ = \ 3 $  One of the above scenarios isn't possible, while the other works.  
November 9th, 2016, 09:44 AM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,031 Thanks: 2342 Math Focus: Mainly analysis and algebra  What follows the quoted line is only true for one particular choice of $t$. It is not in general true that, given two factorizations of an expression into two terms that the factors must be equal.

November 9th, 2016, 02:50 PM  #7  
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  Quote:
Mymathforum.com    11/09/2016 by newuser789 : stripcreator  
November 9th, 2016, 03:14 PM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,031 Thanks: 2342 Math Focus: Mainly analysis and algebra 
I'm not interested in your attempts at humour. If you have something to say to me you can say it here.

November 10th, 2016, 05:05 PM  #9  
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  Quote:
then don't act as if you didn't read it and then make a comment on it to show that you in fact did read it! Maybe next time you will reconsider making a post to someone that is not presumptuous. Hmm? Last edited by Math Message Board tutor; November 10th, 2016 at 05:08 PM.  
November 10th, 2016, 05:37 PM  #10  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,031 Thanks: 2342 Math Focus: Mainly analysis and algebra  Quote:
I still haven't looked at your cartoon because I'm not interested in it. I'm under no obligation to follow and read any links posted by anybody here. If you take issue with the accuracy of my post #6, go ahead and explain what you think is wrong with it. If you have any interesting comment to make on the topic, go ahead an post it. In the absence of the above, I believe this conversation is done.  

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