Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply $\frac{1}{4} (3 \log (x))$ .

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Step 7.2.1

Combine $3$ and $\frac{1}{4}$ .

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

Step 7.2.2

Combine $\frac{3}{4}$ and $\log (x)$ .

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

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Factor $2$ out of $4$ .

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

Step 7.3.2

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

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

Step 7.3.3

Cancel the common factor.

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

Step 7.3.4

Rewrite the expression.

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

Step 7.4

Combine $\frac{1}{2}$ and $\log (y)$ .

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