Precalculus Examples

$\log (\sqrt{2 x^{3} y})$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Apply the distributive property.

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.3

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

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