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 February 2nd, 2011, 06:48 AM #1 Newbie   Joined: Oct 2010 Posts: 6 Thanks: 0 Help solving a problem Hi all, first post and I didn't know exactly where to put this. Does anybody know how to solve for x an equation of the form: 10 + ln(x) = 5x Thanks
 February 2nd, 2011, 09:25 AM #2 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: Help solving a problem You have to use "numerical methods" to get approximate solutions.
 February 2nd, 2011, 10:47 AM #3 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: Help solving a problem If possible, rearrange so that the left-hand side is of the form $f(x)e^{f(x)}$ and the right-hand side is a constant, then take the Lambert W of both sides. $\log \,x= 5x-10\\x = e^{5x-10}\\xe^{-5x}=e^{-10}\\-5xe^{-5x}=\frac{-5}{e^{10}}\\-5x=\operatorname{W}\left(\frac{-5}{e^{10}}\right)\\x=-\frac15 \operatorname{W}\left(\frac{-5}{e^{10}}\right)$ The Lambert W is double-valued over the interval [-1/e, 0). This gives you two solutions, 0.00004541 and 2.15341058.
 February 2nd, 2011, 05:12 PM #4 Newbie   Joined: Oct 2010 Posts: 6 Thanks: 0 Re: Help solving a problem Thanks for the help! Never heard of the Lambert W before.
February 2nd, 2011, 05:14 PM   #5
Global Moderator

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Re: Help solving a problem

Quote:
 Originally Posted by kevinp123 Thanks for the help! Never heard of the Lambert W before.
Me neither, obviously!

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