Algebra Examples

$\log (\frac{x}{\sqrt[3]{y^{4} z^{2}}})$

Step 1

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

$\log (x) - \log (\sqrt[3]{y^{4} z^{2}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{4} z^{2}}$ as ${(y^{4} z^{2})}^{\frac{1}{3}}$ .

$\log (x) - \log ({(y^{4} z^{2})}^{\frac{1}{3}})$

Step 3

Expand $\log ({(y^{4} z^{2})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\log (x) - (\frac{1}{3} \log (y^{4} z^{2}))$

Step 4

Combine $\frac{1}{3}$ and $\log (y^{4} z^{2})$ .

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

Step 5

Rewrite $\log (y^{4} z^{2})$ as $\log (y^{4}) + \log (z^{2})$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

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

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

Step 10

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

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

Simplify the numerator.

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Step 12.1

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

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Multiply $2$ by $- 2$ .

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