Precalculus Examples

$\log ({(\sqrt[3]{x})}^{2} \cdot {(\sqrt[4]{y})}^{7})$

Step 1

Rewrite $\log ({(\sqrt[3]{x})}^{2} \cdot {(\sqrt[4]{y})}^{7})$ as $\log ({(\sqrt[3]{x})}^{2}) + \log ({(\sqrt[4]{y})}^{7})$ .

$\log ({(\sqrt[3]{x})}^{2}) + \log ({(\sqrt[4]{y})}^{7})$

Step 2

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

$\log ({(x^{\frac{1}{3}})}^{2}) + \log ({(\sqrt[4]{y})}^{7})$

Step 3

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

$\log ({(x^{\frac{1}{3}})}^{2}) + \log ({(y^{\frac{1}{4}})}^{7})$

Step 4

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

$2 \log (x^{\frac{1}{3}}) + \log ({(y^{\frac{1}{4}})}^{7})$

Step 5

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

$2 (\frac{1}{3} \log (x)) + \log ({(y^{\frac{1}{4}})}^{7})$

Step 6

Expand $\log ({(y^{\frac{1}{4}})}^{7})$ by moving $7$ outside the logarithm.

$2 (\frac{1}{3} \log (x)) + 7 \log (y^{\frac{1}{4}})$

Step 7

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

$2 (\frac{1}{3} \log (x)) + 7 (\frac{1}{4} \log (y))$

Step 8

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

$\frac{2}{3} \log (x) + 7 (\frac{1}{4} \log (y))$

Step 9

Combine $\frac{1}{4}$ and $7$ .

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

Step 10

Simplify each term.

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Step 10.1

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

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

Step 10.2

Combine $\frac{7}{4}$ and $\log (y)$ .

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