Solve-3-x-4x-

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

S o l v e :

3^{x} = 4 x

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

e^{x l o g 3} = 4 x

e^{- x l o g 3} = \frac{1}{4 x}

- x l o g 3 e^{- x l o g 3} = - \frac{l o g 3}{4}

- x l o g 3 = W_{0} (- \frac{l o g 3}{4})

x = - \frac{W_{0} (- \frac{l o g 3}{4})}{l o g 3} = 0.379 \dots

o r x = 1.794273 \dots

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

W (- \frac{\ln 3}{4}) h a s t w o v a l u e s .

Answered by OlafThorendsen last updated on 15/Jul/20

f (x) = x \ln 3 - \ln x - 2 \ln 2, x > 0

f^{'} (x) = \ln 3 - \frac{1}{x}

f^{'} (x) = 0 \Leftrightarrow x = \frac{1}{\ln 3}

f (\frac{1}{\ln 3}) = 1 + \ln (\ln 3) - 2 \ln 2

f (\frac{1}{\ln 3}) \approx - 0, 29 < 0

\lim_{x \to 0^{-}} f (x) = + \infty

\lim_{x \to + \infty} f (x) = + \infty

\Rightarrow two solutions for f (x) = 0

x \approx 0, 379 or x \approx 1, 794

and f (x) = 0 \Leftrightarrow 3^{x} = 4 x