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人気のある代数 >

5^x-24=(5^2)/5 x

解

5^{x} - 24 = \frac{5 ^{2}}{5} x

解

x = - \frac{5 W _{- 1} ( - \frac{l n ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}} ) + 24 ln ( 5 )}{5 ln ( 5 )}, x = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}} ) + 24 ln ( 5 )}{5 ln ( 5 )}

+1

十進法表記

x = 2.20994 \dots, x = - 4.79991 \dots

解答ステップ

5^{x} - 24 = \frac{5 ^{2}}{5} x

ランベルト形式向けに

5^{x} - 24 = \frac{5 ^{2}}{5} x

を準備する:

1 = (5 x + 24) e^{- l n (5) x}

equationを

(- x - \frac{24}{5}) ln (5) = u

と以下で書き換える:

x = - \frac{5 u + 24 ln ( 5 )}{5 ln ( 5 )}

1 = (5 (- \frac{5 u + 24 ln ( 5 )}{5 ln ( 5 )}) + 24) e^{- l n (5) (- \frac{5 u + 24 l n ( 5 )}{5 l n ( 5 )})}

簡素化

1 = e^{\frac{5 u + 24 l n ( 5 )}{5}} (- \frac{5 u + 24 ln ( 5 )}{ln ( 5 )} + 24)

ランベルト形式で

1 = e^{\frac{5 u + 24 l n ( 5 )}{5}} (- \frac{5 u + 24 ln ( 5 )}{ln ( 5 )} + 24)

を書き換える:

\frac{e ^{\frac{5 u + 24 l n ( 5 )}{5}} u}{e ^{\frac{24 l n ( 5 )}{5}}} = - \frac{ln ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}}

解く

\frac{e ^{\frac{5 u + 24 l n ( 5 )}{5}} u}{e ^{\frac{24 l n ( 5 )}{5}}} = - \frac{ln ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}} : u = W_{- 1} (- \frac{ln ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}}), u = W_{0} (- \frac{ln ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}})

u = W_{- 1} (- \frac{ln ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}}), u = W_{0} (- \frac{ln ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}})

再び

u = (- x - \frac{24}{5}) ln (5)

に置き換えて以下を解く:

x

解く

(- x - \frac{24}{5}) ln (5) = W_{- 1} (- \frac{ln ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}}) : x = - \frac{5 W _{- 1} ( - \frac{l n ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}} ) + 24 ln ( 5 )}{5 ln ( 5 )}

解く

(- x - \frac{24}{5}) ln (5) = W_{0} (- \frac{ln ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}}) : x = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}} ) + 24 ln ( 5 )}{5 ln ( 5 )}

x = - \frac{5 W _{- 1} ( - \frac{l n ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}} ) + 24 ln ( 5 )}{5 ln ( 5 )}, x = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5 e ^{\frac{24 l n ( 5 )}{5}}} ) + 24 ln ( 5 )}{5 ln ( 5 )}

グラフ

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