Precalculus Examples

$\log (\sqrt{x^{3} y^{9} z})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} y^{9} z}$ as ${(x^{3} y^{9} z)}^{\frac{1}{2}}$ .

$\log ({(x^{3} y^{9} z)}^{\frac{1}{2}})$

Step 2

Expand $\log ({(x^{3} y^{9} z)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (x^{3} y^{9} z)$

Step 3

Rewrite $\log (x^{3} y^{9} z)$ as $\log (x^{3} y^{9}) + \log (z)$ .

$\frac{1}{2} (\log (x^{3} y^{9}) + \log (z))$

Step 4

Rewrite $\log (x^{3} y^{9})$ as $\log (x^{3}) + \log (y^{9})$ .

$\frac{1}{2} (\log (x^{3}) + \log (y^{9}) + \log (z))$

Step 5

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

$\frac{1}{2} (3 \log (x) + \log (y^{9}) + \log (z))$

Step 6

Expand $\log (y^{9})$ by moving $9$ outside the logarithm.

$\frac{1}{2} (3 \log (x) + 9 \log (y) + \log (z))$

Step 7

Apply the distributive property.

$\frac{1}{2} (3 \log (x)) + \frac{1}{2} (9 \log (y)) + \frac{1}{2} \log (z)$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{2} (3 \log (x))$ .

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Step 8.1.1

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

$\frac{3}{2} \log (x) + \frac{1}{2} (9 \log (y)) + \frac{1}{2} \log (z)$

Step 8.1.2

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

$\frac{3 \log (x)}{2} + \frac{1}{2} (9 \log (y)) + \frac{1}{2} \log (z)$

Step 8.2

Multiply $\frac{1}{2} (9 \log (y))$ .

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Step 8.2.1

Combine $9$ and $\frac{1}{2}$ .

$\frac{3 \log (x)}{2} + \frac{9}{2} \log (y) + \frac{1}{2} \log (z)$

Step 8.2.2

Combine $\frac{9}{2}$ and $\log (y)$ .

$\frac{3 \log (x)}{2} + \frac{9 \log (y)}{2} + \frac{1}{2} \log (z)$

Step 8.3

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

$\frac{3 \log (x)}{2} + \frac{9 \log (y)}{2} + \frac{\log (z)}{2}$