April 6th, 2017, 08:20 PM  #1 
Senior Member Joined: Jul 2011 Posts: 405 Thanks: 16  exponential equation
number of real solution of $\displaystyle 2017^x2016^x = x$

April 6th, 2017, 08:31 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,549 Thanks: 1399 
$x=0,~x=1$ 2 real solutions 
April 7th, 2017, 07:34 PM  #3 
Senior Member Joined: Jul 2011 Posts: 405 Thanks: 16 
Thanks romsek got it, $\displaystyle \displaystyle \frac{2017^x2016^x}{20172016} = x$ Using LMVT , we have $\displaystyle f'(t) = x, f(t) = t^x\;, t \in \left[2016,2017\right]$ So $\displaystyle xt^{x1} = x$. So we get $\displaystyle x=0\;, 1$ 

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