Expand the Logarithmic Expression log base b of ninth root of (x^4)/(y^9x^5)

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$\log_{b} (\sqrt[9]{\frac{x^{4}}{y^{9} x^{5}}})$

Rewrite $\sqrt[9]{\frac{x^{4}}{y^{9} x^{5}}}$ as ${(\frac{x^{4}}{y^{9} x^{5}})}^{\frac{1}{9}}$ .

$\log_{b} ({(\frac{x^{4}}{y^{9} x^{5}})}^{\frac{1}{9}})$

Expand $\log_{b} ({(\frac{x^{4}}{y^{9} x^{5}})}^{\frac{1}{9}})$ by moving $\frac{1}{9}$ outside the logarithm.

$\frac{1}{9} \log_{b} (\frac{x^{4}}{y^{9} x^{5}})$

Reduce the expression $\frac{x^{4}}{y^{9} x^{5}}$ by cancelling the common factors.

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Multiply by $1$ .

$\frac{1}{9} \log_{b} (\frac{x^{4} \cdot 1}{y^{9} x^{5}})$

Factor $x^{4}$ out of $y^{9} x^{5}$ .

$\frac{1}{9} \log_{b} (\frac{x^{4} \cdot 1}{x^{4} (y^{9} x)})$

Cancel the common factor.

$\frac{1}{9} \log_{b} (\frac{x^{4} \cdot 1}{x^{4} (y^{9} x)})$

Rewrite the expression.

$\frac{1}{9} \log_{b} (\frac{1}{y^{9} x})$

Rewrite $\log_{b} (\frac{1}{y^{9} x})$ as $\log_{b} (1) - \log_{b} (y^{9} x)$ .

$\frac{1}{9} (\log_{b} (1) - \log_{b} (y^{9} x))$

Logarithm base $b$ of $1$ is $0$ .

$\frac{1}{9} (0 - \log_{b} (y^{9} x))$

Subtract $\log_{b} (y^{9} x)$ from $0$ .

$\frac{1}{9} (- \log_{b} (y^{9} x))$

Rewrite $\log_{b} (y^{9} x)$ as $\log_{b} (y^{9}) + \log_{b} (x)$ .

$\frac{1}{9} (- (\log_{b} (y^{9}) + \log_{b} (x)))$

Expand $\log_{b} (y^{9})$ by moving $9$ outside the logarithm.

$\frac{1}{9} (- (9 \log_{b} (y) + \log_{b} (x)))$

Apply the distributive property.

$\frac{1}{9} (- (9 \log_{b} (y)) - \log_{b} (x))$

Multiply $9$ by $- 1$ .

$\frac{1}{9} (- 9 \log_{b} (y) - \log_{b} (x))$

Apply the distributive property.

$\frac{1}{9} (- 9 \log_{b} (y)) + \frac{1}{9} (- \log_{b} (x))$

Cancel the common factor of $9$ .

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Factor $9$ out of $- 9 \log_{b} (y)$ .

$\frac{1}{9} (9 (- \log_{b} (y))) + \frac{1}{9} (- \log_{b} (x))$

Cancel the common factor.

$\frac{1}{9} (9 (- \log_{b} (y))) + \frac{1}{9} (- \log_{b} (x))$

Rewrite the expression.

$- \log_{b} (y) + \frac{1}{9} (- \log_{b} (x))$

Combine $\frac{1}{9}$ and $\log_{b} (x)$ .

$- \log_{b} (y) - \frac{\log_{b} (x)}{9}$

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