Algebra Examples

$\log (\frac{6 {(x + 5)}^{2}}{\sqrt[3]{x^{4}}})$

Step 1

Rewrite $\log (\frac{6 {(x + 5)}^{2}}{\sqrt[3]{x^{4}}})$ as $\log (6 {(x + 5)}^{2}) - \log (\sqrt[3]{x^{4}})$ .

$\log (6 {(x + 5)}^{2}) - \log (\sqrt[3]{x^{4}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{4}}$ as $x^{\frac{4}{3}}$ .

$\log (6 {(x + 5)}^{2}) - \log (x^{\frac{4}{3}})$

Step 3

Expand $\log (x^{\frac{4}{3}})$ by moving $\frac{4}{3}$ outside the logarithm.

$\log (6 {(x + 5)}^{2}) - (\frac{4}{3} \log (x))$

Step 4

Combine $\frac{4}{3}$ and $\log (x)$ .

$\log (6 {(x + 5)}^{2}) - \frac{4 \log (x)}{3}$

Step 5

Rewrite $\log (6 {(x + 5)}^{2})$ as $\log (6) + \log ({(x + 5)}^{2})$ .

$\log (6) + \log ({(x + 5)}^{2}) - \frac{4 \log (x)}{3}$

Step 6

Expand $\log ({(x + 5)}^{2})$ by moving $2$ outside the logarithm.

$\log (6) + 2 \log (x + 5) - \frac{4 \log (x)}{3}$

Step 7

Simplify each term.

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Step 7.1

Rewrite $\log (6)$ as $\log (2 (3))$ .

$\log (2 (3)) + 2 \log (x + 5) - \frac{4 \log (x)}{3}$

Step 7.2

Rewrite $\log (2 (3))$ as $\log (2) + \log (3)$ .

$\log (2) + \log (3) + 2 \log (x + 5) - \frac{4 \log (x)}{3}$

Step 8

To write $2 \log (x + 5)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\log (2) + \log (3) + 2 \log (x + 5) \cdot \frac{3}{3} - \frac{4 \log (x)}{3}$

Step 9

Combine $2 \log (x + 5)$ and $\frac{3}{3}$ .

$\log (2) + \log (3) + \frac{2 \log (x + 5) \cdot 3}{3} - \frac{4 \log (x)}{3}$

Step 10

Combine the numerators over the common denominator.

$\log (2) + \log (3) + \frac{2 \log (x + 5) \cdot 3 - 4 \log (x)}{3}$

Step 11

Simplify the numerator.

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Step 11.1

Factor $2$ out of $2 \log (x + 5) \cdot 3 - 4 \log (x)$ .

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Step 11.1.1

Factor $2$ out of $2 \log (x + 5) \cdot 3$ .

$\log (2) + \log (3) + \frac{2 (\log (x + 5) \cdot 3) - 4 \log (x)}{3}$

Step 11.1.2

Factor $2$ out of $- 4 \log (x)$ .

$\log (2) + \log (3) + \frac{2 (\log (x + 5) \cdot 3) + 2 (- 2 \log (x))}{3}$

Step 11.1.3

Factor $2$ out of $2 (\log (x + 5) \cdot 3) + 2 (- 2 \log (x))$ .

$\log (2) + \log (3) + \frac{2 (\log (x + 5) \cdot 3 - 2 \log (x))}{3}$

Step 11.2

Move $3$ to the left of $\log (x + 5)$ .

$\log (2) + \log (3) + \frac{2 (3 \log (x + 5) - 2 \log (x))}{3}$

Step 12

To write $\log (2)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\log (2) \cdot \frac{3}{3} + \frac{2 (3 \log (x + 5) - 2 \log (x))}{3} + \log (3)$

Step 13

Combine $\log (2)$ and $\frac{3}{3}$ .

$\frac{\log (2) \cdot 3}{3} + \frac{2 (3 \log (x + 5) - 2 \log (x))}{3} + \log (3)$

Step 14

Combine the numerators over the common denominator.

$\frac{\log (2) \cdot 3 + 2 (3 \log (x + 5) - 2 \log (x))}{3} + \log (3)$

Step 15

Simplify the numerator.

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Step 15.1

Move $3$ to the left of $\log (2)$ .

$\frac{3 \cdot \log (2) + 2 (3 \log (x + 5) - 2 \log (x))}{3} + \log (3)$

Step 15.2

Apply the distributive property.

$\frac{3 \log (2) + 2 (3 \log (x + 5)) + 2 (- 2 \log (x))}{3} + \log (3)$

Step 15.3

Multiply $3$ by $2$ .

$\frac{3 \log (2) + 6 \log (x + 5) + 2 (- 2 \log (x))}{3} + \log (3)$

Step 15.4

Multiply $- 2$ by $2$ .

$\frac{3 \log (2) + 6 \log (x + 5) - 4 \log (x)}{3} + \log (3)$

Step 16

Reorder terms.

$\frac{6 \log (x + 5) + 3 \log (2) - 4 \log (x)}{3} + \log (3)$

Step 17

To write $\log (3)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{6 \log (x + 5) + 3 \log (2) - 4 \log (x)}{3} + \log (3) \cdot \frac{3}{3}$

Step 18

Combine $\log (3)$ and $\frac{3}{3}$ .

$\frac{6 \log (x + 5) + 3 \log (2) - 4 \log (x)}{3} + \frac{\log (3) \cdot 3}{3}$

Step 19

Combine the numerators over the common denominator.

$\frac{6 \log (x + 5) + 3 \log (2) - 4 \log (x) + \log (3) \cdot 3}{3}$

Step 20

Move $3$ to the left of $\log (3)$ .

$\frac{6 \log (x + 5) + 3 \log (2) - 4 \log (x) + 3 \log (3)}{3}$