Question Number 103503 by 175mohamed last updated on 15/Jul/20

               Solve :          3^x  = 4x

Answered by Dwaipayan Shikari last updated on 15/Jul/20

e^(xlog3) =4x  e^(−xlog3) =(1/(4x))  −xlog3e^(−xlog3) =−((log3)/4)  −xlog3=W_0 (−((log3)/4))  x=−((W_0 (−((log3)/4)))/(log3))=0.379...  or x=1.794273...

Commented bymr W last updated on 15/Jul/20

W(−((ln 3)/4)) has two values.

Answered by OlafThorendsen last updated on 15/Jul/20

f(x) = xln3−lnx−2ln2, x>0  f′(x) = ln3−(1/x)  f′(x) = 0 ⇔ x = (1/(ln3))  f((1/(ln3))) = 1+ln(ln3)−2ln2  f((1/(ln3))) ≈ −0,29 <0  lim_(x→0^− ) f(x) = +∞  lim_(x→+∞) f(x) = +∞  ⇒ two solutions for f(x) = 0  x ≈ 0,379 or x ≈ 1,794  and f(x) = 0 ⇔ 3^x  = 4x