Solve-x-x-27-x-3-

Question Number 204142 by Tawa11 last updated on 06/Feb/24

Solve : x^{x} = 27^{x - 3}

Answered by Frix last updated on 07/Feb/24

x^{x} = 27^{x - 3}

x \ln x = (x - 3) \ln 27

x (\ln x - \ln 27) = - 3 \ln 3^{3}

x \ln \frac{x}{27} = - 9 \ln 3

[obviously x = 9 because 9 \ln \frac{9}{27} = 9 \ln \frac{1}{3} = - 9 \ln 3]

x = 27 t

27 t \ln t = - 9 \ln 3

t \ln t = - \frac{\ln 3}{3}

t = e^{u}

e^{u} u = - \frac{\ln 3}{3}

u = W (- \frac{\ln 3}{3})

t = e^{W (- \frac{\ln 3}{3})}

x = {27 e}^{W (- \frac{\ln 3}{3})}

x = 9 \lor x \approx 10.8956521444325

Commented by Tawa11 last updated on 07/Feb/24

I appreciate sir .

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