Algebra Examples

{log}_{9} ({(\frac{6^{5}}{(5) (11)})}^{4})

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

Step 1

Expand

{log}_{9} ({(\frac{6^{5}}{(5) (11)})}^{4})

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

4

$4$ outside the logarithm.

4 {log}_{9} (\frac{6^{5}}{(5) (11)})

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

Step 2

Multiply

5

$5$ by

11

$11$ .

4 {log}_{9} (\frac{6^{5}}{55})

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

Step 3

Raise

6

$6$ to the power of

5

$5$ .

4 {log}_{9} (\frac{7776}{55})

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

Step 4

Rewrite

{log}_{9} (\frac{7776}{55})

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

{log}_{9} (7776) - {log}_{9} (55)

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

4 ({log}_{9} (7776) - {log}_{9} (55))

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

Step 5

Rewrite

{log}_{9} (55)

$\log_{9} (55)$ as

{log}_{9} (5 (11))

$\log_{9} (5 (11))$ .

4 ({log}_{9} (7776) - {log}_{9} (5 (11)))

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

Step 6

Rewrite

{log}_{9} (5 (11))

$\log_{9} (5 (11))$ as

{log}_{9} (5) + {log}_{9} (11)

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

4 ({log}_{9} (7776) - ({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))

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

Step 8

Simplify each term.

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Step 8.1

Rewrite

{log}_{9} (7776)

$\log_{9} (7776)$ as

{log}_{9} (2^{5} \cdot 3^{5})

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

4 ({log}_{9} (2^{5} \cdot 3^{5}) - {log}_{9} (5) - {log}_{9} (11))

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

Step 8.2

Rewrite

{log}_{9} (2^{5} \cdot 3^{5})

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

{log}_{9} (2^{5}) + {log}_{9} (3^{5})

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

4 ({log}_{9} (2^{5}) + {log}_{9} (3^{5}) - {log}_{9} (5) - {log}_{9} (11))

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

Step 8.3

Simplify each term.

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Step 8.3.1

Expand

{log}_{9} (2^{5})

$\log_{9} (2^{5})$ by moving

5

$5$ outside the logarithm.

4 (5 {log}_{9} (2) + {log}_{9} (3^{5}) - {log}_{9} (5) - {log}_{9} (11))

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

Step 8.3.2

Expand

{log}_{9} (3^{5})

$\log_{9} (3^{5})$ by moving

5

$5$ outside the logarithm.

4 (5 {log}_{9} (2) + 5 {log}_{9} (3) - {log}_{9} (5) - {log}_{9} (11))

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

Step 8.3.3

Logarithm base

9

$9$ of

3

$3$ is

\frac{1}{2}

$\frac{1}{2}$ .

4 (5 {log}_{9} (2) + 5 (\frac{1}{2}) - {log}_{9} (5) - {log}_{9} (11))

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

Step 8.3.4

Combine

5

$5$ and

\frac{1}{2}

$\frac{1}{2}$ .

4 (5 {log}_{9} (2) + \frac{5}{2} - {log}_{9} (5) - {log}_{9} (11))

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

4 (5 {log}_{9} (2) + \frac{5}{2} - {log}_{9} (5) - {log}_{9} (11))

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

4 (5 {log}_{9} (2) + \frac{5}{2} - {log}_{9} (5) - {log}_{9} (11))

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

Step 9

Apply the distributive property.

4 (5 {log}_{9} (2)) + 4 (\frac{5}{2}) + 4 (- {log}_{9} (5)) + 4 (- {log}_{9} (11))

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

Step 10

Simplify.

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Step 10.1

Multiply

5

$5$ by

4

$4$ .

20 {log}_{9} (2) + 4 (\frac{5}{2}) + 4 (- {log}_{9} (5)) + 4 (- {log}_{9} (11))

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

Step 10.2

Cancel the common factor of

2

$2$ .

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Step 10.2.1

Factor

2

$2$ out of

4

$4$ .

20 {log}_{9} (2) + 2 (2) \frac{5}{2} + 4 (- {log}_{9} (5)) + 4 (- {log}_{9} (11))

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

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

$4$ .

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