Expand the Logarithmic Expression log base 9 of ( sixth root of x^5)/(y^3(z-2)^7)

$\log_{9} (\frac{\sqrt[6]{x^{5}}}{y^{3} {(z - 2)}^{7}})$

Step 1

Rewrite $\log_{9} (\frac{\sqrt[6]{x^{5}}}{y^{3} {(z - 2)}^{7}})$ as $\log_{9} (\sqrt[6]{x^{5}}) - \log_{9} (y^{3} {(z - 2)}^{7})$ .

$\log_{9} (\sqrt[6]{x^{5}}) - \log_{9} (y^{3} {(z - 2)}^{7})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x^{5}}$ as $x^{\frac{5}{6}}$ .

$\log_{9} (x^{\frac{5}{6}}) - \log_{9} (y^{3} {(z - 2)}^{7})$

Step 3

Expand $\log_{9} (x^{\frac{5}{6}})$ by moving $\frac{5}{6}$ outside the logarithm.

$\frac{5}{6} \log_{9} (x) - \log_{9} (y^{3} {(z - 2)}^{7})$

Step 4

Rewrite $\log_{9} (y^{3} {(z - 2)}^{7})$ as $\log_{9} (y^{3}) + \log_{9} ({(z - 2)}^{7})$ .

$\frac{5}{6} \log_{9} (x) - (\log_{9} (y^{3}) + \log_{9} ({(z - 2)}^{7}))$

Step 5

Expand $\log_{9} (y^{3})$ by moving $3$ outside the logarithm.

$\frac{5}{6} \log_{9} (x) - (3 \log_{9} (y) + \log_{9} ({(z - 2)}^{7}))$

Step 6

Expand $\log_{9} ({(z - 2)}^{7})$ by moving $7$ outside the logarithm.

$\frac{5}{6} \log_{9} (x) - (3 \log_{9} (y) + 7 \log_{9} (z - 2))$

Step 7

Simplify each term.

Combine $\frac{5}{6}$ and $\log_{9} (x)$ .

$\frac{5 \log_{9} (x)}{6} - (3 \log_{9} (y) + 7 \log_{9} (z - 2))$

Apply the distributive property.

$\frac{5 \log_{9} (x)}{6} - (3 \log_{9} (y)) - (7 \log_{9} (z - 2))$

Multiply $3$ by $- 1$ .

$\frac{5 \log_{9} (x)}{6} - 3 \log_{9} (y) - (7 \log_{9} (z - 2))$

Multiply $7$ by $- 1$ .

$\frac{5 \log_{9} (x)}{6} - 3 \log_{9} (y) - 7 \log_{9} (z - 2)$

Step 8

To write $- 7 \log_{9} (z - 2)$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5 \log_{9} (x)}{6} - 7 \log_{9} (z - 2) \cdot \frac{6}{6} - 3 \log_{9} (y)$

Step 9

Combine $- 7 \log_{9} (z - 2)$ and $\frac{6}{6}$ .

$\frac{5 \log_{9} (x)}{6} + \frac{- 7 \log_{9} (z - 2) \cdot 6}{6} - 3 \log_{9} (y)$

Step 10

Combine the numerators over the common denominator.

$\frac{5 \log_{9} (x) - 7 \log_{9} (z - 2) \cdot 6}{6} - 3 \log_{9} (y)$

Step 11

Multiply $6$ by $- 7$ .

$\frac{5 \log_{9} (x) - 42 \log_{9} (z - 2)}{6} - 3 \log_{9} (y)$