My Math Forum Solve multiple term exponential

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 March 25th, 2014, 12:11 PM #1 Senior Member   Joined: Sep 2009 Posts: 251 Thanks: 0 Solve multiple term exponential If I take the log of both sides, I get rid of the e on the right, but then I'm stuck with an ln on the left. How do I solve this? $15e^{-.006x}= .005x+1$ What I tried? $e^{-.006x} = \frac{.005x+1}{15} \ln e^{-.006x} = \ln(\frac{.005x+1}{15}) -.006x = \ln(.005x+1)-\ln 15$ Seems no better off than the start. P.S. Calculator not allowed
 March 25th, 2014, 12:34 PM #2 Global Moderator   Joined: May 2007 Posts: 6,834 Thanks: 733 Re: Solve multiple term exponential There is no exact solution.
 March 25th, 2014, 12:37 PM #3 Senior Member   Joined: Sep 2009 Posts: 251 Thanks: 0 Re: Solve multiple term exponential Really? There must be an exact solution.
 March 26th, 2014, 07:11 AM #4 Global Moderator   Joined: Dec 2006 Posts: 21,019 Thanks: 2252 The equation is equivalent to 18e^1.2 = (0.006x + 1.2)e^(0.006x + 1.2), so x = W(18e^1.2)/0.006 - 200, where W denotes the Lambert W function.

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### lambert w sum of exponentials

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