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人気のある代数 >

20x+1=3^x

解

20 x + 1 = 3^{x}

解

x = - \frac{20 W _{- 1} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}, x = - \frac{20 W _{0} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}

+1

十進法表記

x = 4, x = 0

解答ステップ

20 x + 1 = 3^{x}

ランベルト形式向けに

20 x + 1 = 3^{x}

を準備する:

(20 x + 1) e^{- l n (3) x} = 1

equationを

(- x - \frac{1}{20}) ln (3) = u

と以下で書き換える:

x = - \frac{20 u + ln ( 3 )}{20 ln ( 3 )}

(20 (- \frac{20 u + ln ( 3 )}{20 ln ( 3 )}) + 1) e^{- l n (3) (- \frac{20 u + l n ( 3 )}{20 l n ( 3 )})} = 1

簡素化

e^{\frac{20 u + l n ( 3 )}{20}} (- \frac{20 u + ln ( 3 )}{ln ( 3 )} + 1) = 1

ランベルト形式で

e^{\frac{20 u + l n ( 3 )}{20}} (- \frac{20 u + ln ( 3 )}{ln ( 3 )} + 1) = 1

を書き換える:

\frac{e ^{\frac{20 u + l n ( 3 )}{20}} u}{e ^{\frac{l n ( 3 )}{20}}} = - \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}}

解く

\frac{e ^{\frac{20 u + l n ( 3 )}{20}} u}{e ^{\frac{l n ( 3 )}{20}}} = - \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} : u = W_{- 1} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}}), u = W_{0} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}})

u = W_{- 1} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}}), u = W_{0} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}})

再び

u = (- x - \frac{1}{20}) ln (3)

に置き換えて以下を解く:

x

解く

(- x - \frac{1}{20}) ln (3) = W_{- 1} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}}) : x = - \frac{20 W _{- 1} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}

解く

(- x - \frac{1}{20}) ln (3) = W_{0} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}}) : x = - \frac{20 W _{0} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}

x = - \frac{20 W _{- 1} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}, x = - \frac{20 W _{0} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}

グラフ

人気の例

(\frac{1}{5})^{x} = 625

5(2^{2p+1})-17(2^{p+3})=0

5 (2^{2 p + 1}) - 17 (2^{p + 3}) = 0

2048 = 4 \cdot 2^{x}

2^{2x-1}-5*2^{x-1}+2=0

2^{2 x - 1} - 5 \cdot 2^{x - 1} + 2 = 0

(5/3)^{4x-12}=1

(\frac{5}{3})^{4 x - 12} = 1