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

               Solve :          3^x  = 4x

$$\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:{Solve}\:: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{3}^{{x}} \:=\:\mathrm{4}{x} \\ $$

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...

$${e}^{{xlog}\mathrm{3}} =\mathrm{4}{x} \\ $$$${e}^{−{xlog}\mathrm{3}} =\frac{\mathrm{1}}{\mathrm{4}{x}} \\ $$$$−{xlog}\mathrm{3}{e}^{−{xlog}\mathrm{3}} =−\frac{{log}\mathrm{3}}{\mathrm{4}} \\ $$$$−{xlog}\mathrm{3}={W}_{\mathrm{0}} \left(−\frac{{log}\mathrm{3}}{\mathrm{4}}\right) \\ $$$${x}=−\frac{{W}_{\mathrm{0}} \left(−\frac{{log}\mathrm{3}}{\mathrm{4}}\right)}{{log}\mathrm{3}}=\mathrm{0}.\mathrm{379}... \\ $$$${or}\:{x}=\mathrm{1}.\mathrm{794273}... \\ $$

Commented by mr W last updated on 15/Jul/20

W(−((ln 3)/4)) has two values.

$${W}\left(−\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{4}}\right)\:{has}\:{two}\:{values}. \\ $$

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

$${f}\left({x}\right)\:=\:{x}\mathrm{ln3}−\mathrm{ln}{x}−\mathrm{2ln2},\:{x}>\mathrm{0} \\ $$$${f}'\left({x}\right)\:=\:\mathrm{ln3}−\frac{\mathrm{1}}{{x}} \\ $$$${f}'\left({x}\right)\:=\:\mathrm{0}\:\Leftrightarrow\:{x}\:=\:\frac{\mathrm{1}}{\mathrm{ln3}} \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{ln3}}\right)\:=\:\mathrm{1}+\mathrm{ln}\left(\mathrm{ln3}\right)−\mathrm{2ln2} \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{ln3}}\right)\:\approx\:−\mathrm{0},\mathrm{29}\:<\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}{f}\left({x}\right)\:=\:+\infty \\ $$$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}{f}\left({x}\right)\:=\:+\infty \\ $$$$\Rightarrow\:\mathrm{two}\:\mathrm{solutions}\:\mathrm{for}\:{f}\left({x}\right)\:=\:\mathrm{0} \\ $$$${x}\:\approx\:\mathrm{0},\mathrm{379}\:\mathrm{or}\:{x}\:\approx\:\mathrm{1},\mathrm{794} \\ $$$$\mathrm{and}\:{f}\left({x}\right)\:=\:\mathrm{0}\:\Leftrightarrow\:\mathrm{3}^{{x}} \:=\:\mathrm{4}{x} \\ $$

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