Posted 25 Mar 2013 There are three solutions. Can you find a formal proof for all three? 0 Share this post Link to post Share on other sites

0 Posted 25 Mar 2013 x=2 and 4. Proof: 2^{2} = 2^{2} 4^{2} = (2^{2})^{2} = 2^{4}. The third solution is x = -.76666468...., found numerically, looking for closed form. 0 Share this post Link to post Share on other sites

0 Posted 25 Mar 2013 Using natural logarithms, the equation becomesx = 2 ln(x) x = 2 = 2 ln (2) = 2 x 1 = 2 x = 4 = 2 ln (4) = 2 x 2 = 4 The third root is negative and solvesx = -2 ln (x) Which has no algebraic solution. Numerically, it'sx = -0.766664695 .... Interested to find out what you mean by formal proof. 0 Share this post Link to post Share on other sites

0 Posted 25 Mar 2013 roots are easy to find and 2 roots may be somehow proved. 3rd is difficult to prove. Using log base e, 2 lnx = x ln 2 x/lnx = 2/ln2, By comparison, x=2 and also LHS= 4/2lnx, again compare, x=4. 0 Share this post Link to post Share on other sites

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There are three solutions. Can you find a formal proof for all three?

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