簡約 log_{5}(2^{-log_{2}(5)})

解

簡約

lo g_{5} (2^{- l o g_{2} (5)})

- 1

解答ステップ

lo g_{5} (2^{- l o g_{2} (5)})

対数の規則を適用する:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x), x > 0

lo g_{5} (2^{- l o g_{2} (5)}) = - lo g_{2} (5) lo g_{5} (2)

= - lo g_{2} (5) lo g_{5} (2)

対数の規則を適用する:

lo g_{a} (b) = \frac{lo g _{c} ( b )}{lo g _{c} ( a )}

lo g_{2} (5) = \frac{ln ( 5 )}{ln ( 2 )}

= - \frac{ln ( 5 )}{ln ( 2 )} lo g_{5} (2)

対数の規則を適用する:

lo g_{a} (b) = \frac{lo g _{c} ( b )}{lo g _{c} ( a )}

lo g_{5} (2) = \frac{ln ( 2 )}{ln ( 5 )}

= - \frac{ln ( 5 )}{ln ( 2 )} \cdot \frac{ln ( 2 )}{ln ( 5 )}

クロス約分する:

\frac{a}{b} \cdot \frac{b}{a} = 1

\frac{ln ( 5 )}{ln ( 2 )} \cdot \frac{ln ( 2 )}{ln ( 5 )} = 1

= - 1