Precalculus Examples

$\log (\frac{\sqrt[3]{x^{7} z}}{y^{2}})$

Step 1

Rewrite $\log (\frac{\sqrt[3]{x^{7} z}}{y^{2}})$ as $\log (\sqrt[3]{x^{7} z}) - \log (y^{2})$ .

$\log (\sqrt[3]{x^{7} z}) - \log (y^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{7} z}$ as ${(x^{7} z)}^{\frac{1}{3}}$ .

$\log ({(x^{7} z)}^{\frac{1}{3}}) - \log (y^{2})$

Step 3

Expand $\log ({(x^{7} z)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log (x^{7} z) - \log (y^{2})$

Step 4

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

$\frac{1}{3} \log (x^{7} z) - (2 \log (y))$

Step 5

Multiply $2$ by $- 1$ .

$\frac{1}{3} \log (x^{7} z) - 2 \log (y)$

Step 6

Rewrite $\log (x^{7} z)$ as $\log (x^{7}) + \log (z)$ .

$\frac{1}{3} (\log (x^{7}) + \log (z)) - 2 \log (y)$

Step 7

Expand $\log (x^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{3} (7 \log (x) + \log (z)) - 2 \log (y)$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$\frac{1}{3} (7 \log (x)) + \frac{1}{3} \log (z) - 2 \log (y)$

Step 8.2

Multiply $\frac{1}{3} (7 \log (x))$ .

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Step 8.2.1

Combine $7$ and $\frac{1}{3}$ .

$\frac{7}{3} \log (x) + \frac{1}{3} \log (z) - 2 \log (y)$

Step 8.2.2

Combine $\frac{7}{3}$ and $\log (x)$ .

$\frac{7 \log (x)}{3} + \frac{1}{3} \log (z) - 2 \log (y)$

Step 8.3

Combine $\frac{1}{3}$ and $\log (z)$ .

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