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expandir log_{4}(\sqrt[9]{((x^2y))/(16)})

Solução

expandir

lo g_{4} (9 \frac{( x ^{2} y )}{16})

- \frac{2}{9} + \frac{lo g _{4} ( y )}{9} + \frac{2 lo g _{4} ( x )}{9}

Passos da solução

lo g_{4} (9 \frac{x ^{2} y}{16})

Reescrever como

= lo g_{4} ((\frac{x ^{2} y}{16})^{\frac{1}{9}})

Aplicar as propriedades dos logaritmos

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

assumindo que

x \geq 0

= \frac{1}{9} lo g_{4} (\frac{x ^{2} y}{16})

Simplificar

lo g_{4} (\frac{x ^{2} y}{16}) : 2 lo g_{4} (x) + lo g_{4} (y) - 2

= \frac{1}{9} (2 lo g_{4} (x) + lo g_{4} (y) - 2)

Aplicar a seguinte regra dos produtos notáveis

= \frac{1}{9} lo g_{4} (y) + \frac{1}{9} \cdot 2 lo g_{4} (x) + \frac{1}{9} (- 2)

Aplicar as regras dos sinais

+ (- a) = - a

= \frac{1}{9} lo g_{4} (y) + 2 \cdot \frac{1}{9} lo g_{4} (x) - 2 \cdot \frac{1}{9}

2 \cdot \frac{1}{9} lo g_{4} (x) = \frac{2}{9} lo g_{4} (x)

2 \cdot \frac{1}{9} = \frac{2}{9}

= \frac{1}{9} lo g_{4} (y) + \frac{2}{9} lo g_{4} (x) - \frac{2}{9}

Multiplicar

\frac{1}{9} lo g_{4} (y) : \frac{lo g _{4} ( y )}{9}

= \frac{lo g _{4} ( y )}{9} + lo g_{4} (x) \frac{2}{9} - \frac{2}{9}

Multiplicar

lo g_{4} (x) \frac{2}{9} : \frac{2 lo g _{4} ( x )}{9}

= \frac{lo g _{4} ( y )}{9} + \frac{2 lo g _{4} ( x )}{9} - \frac{2}{9}

Combinar as frações usando o mínimo múltiplo comum:

\frac{lo g _{4} ( y ) + 2 lo g _{4} ( x )}{9}

= - \frac{2}{9} + \frac{2 lo g _{4} ( x ) + lo g _{4} ( y )}{9}

Aplicar a regra

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 2 + lo g _{4} ( y ) + 2 lo g _{4} ( x )}{9}

- 2 + lo g_{4} (y) + 2 lo g_{4} (x) = - 2 + lo g_{4} (y x^{2})

= \frac{- 2 + lo g _{4} ( x ^{2} y )}{9}

lo g_{4} (y x^{2}) = lo g_{4} (y) + 2 lo g_{4} (x)

= \frac{- 2 + lo g _{4} ( y ) + 2 lo g _{4} ( x )}{9}

Aplicar as propriedades das frações:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 2 + lo g _{4} ( y ) + 2 lo g _{4} ( x )}{9} = - \frac{2}{9} + \frac{lo g _{4} ( y )}{9} + \frac{2 lo g _{4} ( x )}{9}

= - \frac{2}{9} + \frac{lo g _{4} ( y )}{9} + \frac{2 lo g _{4} ( x )}{9}

s im pl i f y b (6 - \frac{b ^{2}}{4})

e x p an d b (y + b - x)^{2}

e x p an d \frac{5 ( 8 - 19 )}{6}

e x p an d (2 + \frac{1}{3})^{3}

e x p an d (2 cos (2 t) + 1 + 4 sin (2 t))^{2}