Expand the Logarithmic Expression log of x^2y^3 cube root of x^2y^5

$\log (x^{2} y^{3} \sqrt[3]{x^{2} y^{5}})$

Step 1

Rewrite $\log (x^{2} y^{3} \sqrt[3]{x^{2} y^{5}})$ as $\log (x^{2} y^{3}) + \log (\sqrt[3]{x^{2} y^{5}})$ .

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

Step 2

Rewrite $\log (x^{2} y^{3})$ as $\log (x^{2}) + \log (y^{3})$ .

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

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2} y^{5}}$ as ${(x^{2} y^{5})}^{\frac{1}{3}}$ .

$\log (x^{2}) + \log (y^{3}) + \log ({(x^{2} y^{5})}^{\frac{1}{3}})$

Step 4

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

$2 \log (x) + \log (y^{3}) + \log ({(x^{2} y^{5})}^{\frac{1}{3}})$

Step 5

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

$2 \log (x) + 3 \log (y) + \log ({(x^{2} y^{5})}^{\frac{1}{3}})$

Step 6

Expand $\log ({(x^{2} y^{5})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$2 \log (x) + 3 \log (y) + \frac{1}{3} \log (x^{2} y^{5})$

Step 7

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

$2 \log (x) + 3 \log (y) + \frac{1}{3} (\log (x^{2}) + \log (y^{5}))$

Step 8

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

$2 \log (x) + 3 \log (y) + \frac{1}{3} (2 \log (x) + \log (y^{5}))$

Step 9

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

$2 \log (x) + 3 \log (y) + \frac{1}{3} (2 \log (x) + 5 \log (y))$

Step 10

Simplify each term.

Apply the distributive property.

$2 \log (x) + 3 \log (y) + \frac{1}{3} (2 \log (x)) + \frac{1}{3} (5 \log (y))$

Multiply $\frac{1}{3} (2 \log (x))$ .

Combine $2$ and $\frac{1}{3}$ .

$2 \log (x) + 3 \log (y) + \frac{2}{3} \log (x) + \frac{1}{3} (5 \log (y))$

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

$2 \log (x) + 3 \log (y) + \frac{2 \log (x)}{3} + \frac{1}{3} (5 \log (y))$

Multiply $\frac{1}{3} (5 \log (y))$ .

Combine $5$ and $\frac{1}{3}$ .

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

Combine $\frac{5}{3}$ and $\log (y)$ .

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

Step 11

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

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

Step 12

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

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

Step 13

Combine the numerators over the common denominator.

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

Step 14

Simplify the numerator.

Multiply $3$ by $2$ .

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

Add $6 \log (x)$ and $2 \log (x)$ .

$3 \log (y) + \frac{8 \log (x)}{3} + \frac{5 \log (y)}{3}$

Step 15

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

$3 \log (y) \cdot \frac{3}{3} + \frac{5 \log (y)}{3} + \frac{8 \log (x)}{3}$

Step 16

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

$\frac{3 \log (y) \cdot 3}{3} + \frac{5 \log (y)}{3} + \frac{8 \log (x)}{3}$

Step 17

Combine the numerators over the common denominator.

$\frac{3 \log (y) \cdot 3 + 5 \log (y)}{3} + \frac{8 \log (x)}{3}$

Step 18

Combine the numerators over the common denominator.

$\frac{3 \log (y) \cdot 3 + 5 \log (y) + 8 \log (x)}{3}$

Step 19

Simplify the numerator.

Multiply $3$ by $3$ .

$\frac{9 \log (y) + 5 \log (y) + 8 \log (x)}{3}$

Add $9 \log (y)$ and $5 \log (y)$ .

$\frac{14 \log (y) + 8 \log (x)}{3}$

Step 20

Factor $2$ out of $14 \log (y) + 8 \log (x)$ .

Factor $2$ out of $14 \log (y)$ .

$\frac{2 (7 \log (y)) + 8 \log (x)}{3}$

Factor $2$ out of $8 \log (x)$ .

$\frac{2 (7 \log (y)) + 2 (4 \log (x))}{3}$

Factor $2$ out of $2 (7 \log (y)) + 2 (4 \log (x))$ .

$\frac{2 (7 \log (y) + 4 \log (x))}{3}$

Expand the Logarithmic Expression log of x^2y^3 cube root of x^2y^5

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