Algebra Examples

$\log_{9} ({(\frac{6^{5}}{(5) (11)})}^{4})$

Step 1

Expand $\log_{9} ({(\frac{6^{5}}{(5) (11)})}^{4})$ by moving $4$ outside the logarithm.

$4 \log_{9} (\frac{6^{5}}{(5) (11)})$

Step 2

Multiply $5$ by $11$ .

$4 \log_{9} (\frac{6^{5}}{55})$

Step 3

Raise $6$ to the power of $5$ .

$4 \log_{9} (\frac{7776}{55})$

Step 4

Rewrite $\log_{9} (\frac{7776}{55})$ as $\log_{9} (7776) - \log_{9} (55)$ .

$4 (\log_{9} (7776) - \log_{9} (55))$

Step 5

Rewrite $\log_{9} (55)$ as $\log_{9} (5 (11))$ .

$4 (\log_{9} (7776) - \log_{9} (5 (11)))$

Step 6

Rewrite $\log_{9} (5 (11))$ as $\log_{9} (5) + \log_{9} (11)$ .

$4 (\log_{9} (7776) - (\log_{9} (5) + \log_{9} (11)))$

Step 7

Apply the distributive property.

$4 (\log_{9} (7776) - \log_{9} (5) - \log_{9} (11))$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\log_{9} (7776)$ as $\log_{9} (2^{5} \cdot 3^{5})$ .

$4 (\log_{9} (2^{5} \cdot 3^{5}) - \log_{9} (5) - \log_{9} (11))$

Step 8.2

Rewrite $\log_{9} (2^{5} \cdot 3^{5})$ as $\log_{9} (2^{5}) + \log_{9} (3^{5})$ .

$4 (\log_{9} (2^{5}) + \log_{9} (3^{5}) - \log_{9} (5) - \log_{9} (11))$

Step 8.3

Simplify each term.

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Step 8.3.1

Expand $\log_{9} (2^{5})$ by moving $5$ outside the logarithm.

$4 (5 \log_{9} (2) + \log_{9} (3^{5}) - \log_{9} (5) - \log_{9} (11))$

Step 8.3.2

Expand $\log_{9} (3^{5})$ by moving $5$ outside the logarithm.

$4 (5 \log_{9} (2) + 5 \log_{9} (3) - \log_{9} (5) - \log_{9} (11))$

Step 8.3.3

Logarithm base $9$ of $3$ is $\frac{1}{2}$ .

$4 (5 \log_{9} (2) + 5 (\frac{1}{2}) - \log_{9} (5) - \log_{9} (11))$

Step 8.3.4

Combine $5$ and $\frac{1}{2}$ .

$4 (5 \log_{9} (2) + \frac{5}{2} - \log_{9} (5) - \log_{9} (11))$

Step 9

Apply the distributive property.

$4 (5 \log_{9} (2)) + 4 (\frac{5}{2}) + 4 (- \log_{9} (5)) + 4 (- \log_{9} (11))$

Step 10

Simplify.

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Step 10.1

Multiply $5$ by $4$ .

$20 \log_{9} (2) + 4 (\frac{5}{2}) + 4 (- \log_{9} (5)) + 4 (- \log_{9} (11))$

Step 10.2

Cancel the common factor of $2$ .

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Step 10.2.1

Factor $2$ out of $4$ .

$20 \log_{9} (2) + 2 (2) \frac{5}{2} + 4 (- \log_{9} (5)) + 4 (- \log_{9} (11))$

Step 10.2.2

Cancel the common factor.

$20 \log_{9} (2) + 2 \cdot 2 \frac{5}{2} + 4 (- \log_{9} (5)) + 4 (- \log_{9} (11))$

Step 10.2.3

Rewrite the expression.

$20 \log_{9} (2) + 2 \cdot 5 + 4 (- \log_{9} (5)) + 4 (- \log_{9} (11))$

Step 10.3

Multiply $2$ by $5$ .

$20 \log_{9} (2) + 10 + 4 (- \log_{9} (5)) + 4 (- \log_{9} (11))$

Step 10.4

Multiply $- 1$ by $4$ .

$20 \log_{9} (2) + 10 - 4 \log_{9} (5) + 4 (- \log_{9} (11))$

Step 10.5

Multiply $- 1$ by $4$ .

$20 \log_{9} (2) + 10 - 4 \log_{9} (5) - 4 \log_{9} (11)$