Precalculus Examples

$\log (\frac{x^{5}}{\sqrt[3]{z^{8} y^{2}}})$

Step 1

Rewrite $\log (\frac{x^{5}}{\sqrt[3]{z^{8} y^{2}}})$ as $\log (x^{5}) - \log (\sqrt[3]{z^{8} y^{2}})$ .

$\log (x^{5}) - \log (\sqrt[3]{z^{8} y^{2}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{z^{8} y^{2}}$ as ${(z^{8} y^{2})}^{\frac{1}{3}}$ .

$\log (x^{5}) - \log ({(z^{8} y^{2})}^{\frac{1}{3}})$

Step 3

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

$5 \log (x) - \log ({(z^{8} y^{2})}^{\frac{1}{3}})$

Step 4

Expand $\log ({(z^{8} y^{2})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$5 \log (x) - (\frac{1}{3} \log (z^{8} y^{2}))$

Step 5

Combine $\frac{1}{3}$ and $\log (z^{8} y^{2})$ .

$5 \log (x) - \frac{\log (z^{8} y^{2})}{3}$

Step 6

Rewrite $\log (z^{8} y^{2})$ as $\log (z^{8}) + \log (y^{2})$ .

$5 \log (x) - \frac{\log (z^{8}) + \log (y^{2})}{3}$

Step 7

Expand $\log (z^{8})$ by moving $8$ outside the logarithm.

$5 \log (x) - \frac{8 \log (z) + \log (y^{2})}{3}$

Step 8

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

$5 \log (x) - \frac{8 \log (z) + 2 \log (y)}{3}$

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

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

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

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Simplify the numerator.

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Step 13.1

Multiply $3$ by $5$ .

$\frac{15 \log (x) - 2 (4 \log (z) + \log (y))}{3}$

Step 13.2

Apply the distributive property.

$\frac{15 \log (x) - 2 (4 \log (z)) - 2 \log (y)}{3}$

Step 13.3

Multiply $4$ by $- 2$ .

$\frac{15 \log (x) - 8 \log (z) - 2 \log (y)}{3}$