What is expand log_{3}(\sqrt[9]{(x^2y)/9}) ?

The solution to expand log_{3}(\sqrt[9]{(x^2y)/9}) is -2/9+(log_{3}(y))/9+(2log_{3}(x))/9

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expand log_{3}(\sqrt[9]{(x^2y)/9})

expand

- \frac{2}{9} + \frac{lo g _{3} ( y )}{9} + \frac{2 lo g _{3} ( x )}{9}

Solution steps

Rewrite as

= lo g_{3} ((\frac{x ^{2} y}{9})^{\frac{1}{9}})

Apply log rule

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

assuming

x \geq 0

= \frac{1}{9} lo g_{3} (\frac{x ^{2} y}{9})

Simplify

lo g_{3} (\frac{x ^{2} y}{9}) : 2 lo g_{3} (x) + lo g_{3} (y) - 2

= \frac{1}{9} (2 lo g_{3} (x) + lo g_{3} (y) - 2)

Distribute parentheses

= \frac{1}{9} lo g_{3} (y) + \frac{1}{9} \cdot 2 lo g_{3} (x) + \frac{1}{9} (- 2)

Apply minus-plus rules

+ (- a) = - a

= \frac{1}{9} lo g_{3} (y) + 2 \cdot \frac{1}{9} lo g_{3} (x) - 2 \cdot \frac{1}{9}

2 \cdot \frac{1}{9} lo g_{3} (x) = \frac{2}{9} lo g_{3} (x)

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

= \frac{1}{9} lo g_{3} (y) + \frac{2}{9} lo g_{3} (x) - \frac{2}{9}

Multiply

\frac{1}{9} lo g_{3} (y) : \frac{lo g _{3} ( y )}{9}

= \frac{lo g _{3} ( y )}{9} + lo g_{3} (x) \frac{2}{9} - \frac{2}{9}

Multiply

lo g_{3} (x) \frac{2}{9} : \frac{2 lo g _{3} ( x )}{9}

= \frac{lo g _{3} ( y )}{9} + \frac{2 lo g _{3} ( x )}{9} - \frac{2}{9}

Combine the fractions

\frac{lo g _{3} ( y )}{9} + \frac{2 lo g _{3} ( x )}{9} : \frac{lo g _{3} ( y ) + 2 lo g _{3} ( x )}{9}

= - \frac{2}{9} + \frac{2 lo g _{3} ( x ) + lo g _{3} ( y )}{9}

Apply rule

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

= \frac{- 2 + lo g _{3} ( y ) + 2 lo g _{3} ( x )}{9}

- 2 + lo g_{3} (y) + 2 lo g_{3} (x) = - 2 + lo g_{3} (y x^{2})

= \frac{- 2 + lo g _{3} ( x ^{2} y )}{9}

lo g_{3} (y x^{2}) = lo g_{3} (y) + 2 lo g_{3} (x)

= \frac{- 2 + lo g _{3} ( y ) + 2 lo g _{3} ( x )}{9}

Apply the fraction rule:

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

\frac{- 2 + lo g _{3} ( y ) + 2 lo g _{3} ( x )}{9} = - \frac{2}{9} + \frac{lo g _{3} ( y )}{9} + \frac{2 lo g _{3} ( x )}{9}

= - \frac{2}{9} + \frac{lo g _{3} ( y )}{9} + \frac{2 lo g _{3} ( x )}{9}

cos (x) (x - 1)

\frac{( 2 s + 1 ) e ^{- 4 s}}{4 s ^{2} + 4 s + 5}

\frac{d}{d x} ((x + a) e^{2 x}) + b

3 x \cdot (5 + b x + 5)

D x (x) (2 x + 4)

What is expand log_{3}(\sqrt[9]{(x^2y)/9}) ?
The solution to expand log_{3}(\sqrt[9]{(x^2y)/9}) is -2/9+(log_{3}(y))/9+(2log_{3}(x))/9