Precalculus Examples

$\log (\sqrt{x^{5} y z^{3}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{5} y z^{3}}$ as ${(x^{5} y z^{3})}^{\frac{1}{2}}$ .

$\log ({(x^{5} y z^{3})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(x^{5} y z^{3})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (x^{5} y z^{3})$

Step 3

Rewrite $\log (x^{5} y z^{3})$ as $\log (x^{5} y) + \log (z^{3})$ .

$\frac{1}{2} (\log (x^{5} y) + \log (z^{3}))$

Step 4

Rewrite $\log (x^{5} y)$ as $\log (x^{5}) + \log (y)$ .

$\frac{1}{2} (\log (x^{5}) + \log (y) + \log (z^{3}))$

Step 5

Expand $\log (x^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{2} (5 \log (x) + \log (y) + \log (z^{3}))$

Step 6

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

$\frac{1}{2} (5 \log (x) + \log (y) + 3 \log (z))$

Step 7

Apply the distributive property.

$\frac{1}{2} (5 \log (x)) + \frac{1}{2} \log (y) + \frac{1}{2} (3 \log (z))$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{2} (5 \log (x))$ .

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Step 8.1.1

Combine $5$ and $\frac{1}{2}$ .

$\frac{5}{2} \log (x) + \frac{1}{2} \log (y) + \frac{1}{2} (3 \log (z))$

Step 8.1.2

Combine $\frac{5}{2}$ and $\log (x)$ .

$\frac{5 \log (x)}{2} + \frac{1}{2} \log (y) + \frac{1}{2} (3 \log (z))$

Step 8.2

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

$\frac{5 \log (x)}{2} + \frac{\log (y)}{2} + \frac{1}{2} (3 \log (z))$

Step 8.3

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

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Step 8.3.1

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

$\frac{5 \log (x)}{2} + \frac{\log (y)}{2} + \frac{3}{2} \log (z)$

Step 8.3.2

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

$\frac{5 \log (x)}{2} + \frac{\log (y)}{2} + \frac{3 \log (z)}{2}$