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Beliebt Algebra >

2^2x+3+7=15.2^x

Lösung

2^{2} x + 3 + 7 = 15. 2^{x}

Lösung

x = - \frac{2 W _{- 1} ( - \frac{l n ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}} ) + 5 ln ( \frac{76}{5} )}{2 ln ( \frac{76}{5} )}, x = - \frac{2 W _{0} ( - \frac{l n ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}} ) + 5 ln ( \frac{76}{5} )}{2 ln ( \frac{76}{5} )}

+1

Dezimale

x = 0.96621 \dots, x = - 2.49972 \dots

Schritte zur Lösung

2^{2} x + 3 + 7 = 15. 2^{x}

2^{2} x + 3 + 7 = 15. 2^{x}

für die Lambert-Form vorbereiten:

e^{- l n (\frac{76}{5}) x} (4 x + 10) = 1

Schreibe die Gleichung um mit

ln (\frac{76}{5}) (- x - \frac{5}{2}) = u

und

x = - \frac{2 u + 5 ln ( \frac{76}{5} )}{2 ln ( \frac{76}{5} )}

e^{- l n (\frac{76}{5}) (- \frac{2 u + 5 l n ( \frac{76}{5} )}{2 l n ( \frac{76}{5} )})} (4 (- \frac{2 u + 5 ln ( \frac{76}{5} )}{2 ln ( \frac{76}{5} )}) + 10) = 1

Vereinfache

e^{\frac{2 u + 5 l n ( \frac{76}{5} )}{2}} (- \frac{2 ( 2 u + 5 ln ( \frac{76}{5} ) )}{ln ( \frac{76}{5} )} + 10) = 1

e^{\frac{2 u + 5 l n ( \frac{76}{5} )}{2}} (- \frac{2 ( 2 u + 5 ln ( \frac{76}{5} ) )}{ln ( \frac{76}{5} )} + 10) = 1

in Lambert-Form umschreiben:

\frac{e ^{\frac{2 u + 5 l n ( \frac{76}{5} )}{2}} u}{e ^{\frac{5 l n ( \frac{76}{5} )}{2}}} = - \frac{ln ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}}

Löse

\frac{e ^{\frac{2 u + 5 l n ( \frac{76}{5} )}{2}} u}{e ^{\frac{5 l n ( \frac{76}{5} )}{2}}} = - \frac{ln ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}} : u = W_{- 1} - \frac{ln ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}}, u = W_{0} - \frac{ln ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}}

u = W_{- 1} - \frac{ln ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}}, u = W_{0} - \frac{ln ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}}

Setze

u = ln (\frac{76}{5}) (- x - \frac{5}{2}) w i e d er e in,

löse für

x

Löse

ln (\frac{76}{5}) (- x - \frac{5}{2}) = W_{- 1} - \frac{ln ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}} : x = - \frac{2 W _{- 1} ( - \frac{l n ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}} ) + 5 ln ( \frac{76}{5} )}{2 ln ( \frac{76}{5} )}

Löse

ln (\frac{76}{5}) (- x - \frac{5}{2}) = W_{0} - \frac{ln ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}} : x = - \frac{2 W _{0} ( - \frac{l n ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}} ) + 5 ln ( \frac{76}{5} )}{2 ln ( \frac{76}{5} )}

x = - \frac{2 W _{- 1} ( - \frac{l n ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}} ) + 5 ln ( \frac{76}{5} )}{2 ln ( \frac{76}{5} )}, x = - \frac{2 W _{0} ( - \frac{l n ( \frac{76}{5} )}{4 e ^{\frac{5 l n ( \frac{76}{5} )}{2}}} ) + 5 ln ( \frac{76}{5} )}{2 ln ( \frac{76}{5} )}

Graph

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