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人気のある代数 >

2^x+1=2x+5

解

2^{x} + 1 = 2 x + 5

解

x = - \frac{W _{- 1} ( - \frac{l n ( 2 )}{8} ) + 2 ln ( 2 )}{ln ( 2 )}, x = - \frac{W _{0} ( - \frac{l n ( 2 )}{8} ) + 2 ln ( 2 )}{ln ( 2 )}

+1

十進法表記

x = 3.44490 \dots, x = - 1.86250 \dots

解答ステップ

2^{x} + 1 = 2 x + 5

ランベルト形式向けに

2^{x} + 1 = 2 x + 5

を準備する:

1 = (2 x + 4) e^{- l n (2) x}

equationを

(- x - 2) ln (2) = u

と以下で書き換える:

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

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

簡素化

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

ランベルト形式で

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

を書き換える:

e^{u} u = - \frac{ln ( 2 )}{8}

解く

e^{u} u = - \frac{ln ( 2 )}{8} : u = W_{- 1} (- \frac{ln ( 2 )}{8}), u = W_{0} (- \frac{ln ( 2 )}{8})

u = W_{- 1} (- \frac{ln ( 2 )}{8}), u = W_{0} (- \frac{ln ( 2 )}{8})

再び

u = (- x - 2) ln (2)

に置き換えて以下を解く:

x

解く

(- x - 2) ln (2) = W_{- 1} (- \frac{ln ( 2 )}{8}) : x = - \frac{W _{- 1} ( - \frac{l n ( 2 )}{8} ) + 2 ln ( 2 )}{ln ( 2 )}

解く

(- x - 2) ln (2) = W_{0} (- \frac{ln ( 2 )}{8}) : x = - \frac{W _{0} ( - \frac{l n ( 2 )}{8} ) + 2 ln ( 2 )}{ln ( 2 )}

x = - \frac{W _{- 1} ( - \frac{l n ( 2 )}{8} ) + 2 ln ( 2 )}{ln ( 2 )}, x = - \frac{W _{0} ( - \frac{l n ( 2 )}{8} ) + 2 ln ( 2 )}{ln ( 2 )}

グラフ

人気の例

16^x= 1/(sqrt(2))

1 6^{x} = \frac{1}{2}

45 = 3 0^{x^{6}}

6^{5x+1}=6^{7x-1}

6^{5 x + 1} = 6^{7 x - 1}

3^{4 x + 3} = 8 1^{x}

solvefor k,15=1+19e^k

so l v e f or k, 15 = 1 + 19 e^{k}