Precalculus Examples

$\log (5 \sqrt{12})$

Step 1

Rewrite $\log (5 \sqrt{12})$ as $\log (5) + \log (\sqrt{12})$ .

$\log (5) + \log (\sqrt{12})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{12}$ as $12^{\frac{1}{2}}$ .

$\log (5) + \log (12^{\frac{1}{2}})$

Step 3

Expand $\log (12^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log (5) + \frac{1}{2} \log (12)$

Step 4

Rewrite $\log (12)$ as $\log (2^{2} \cdot 3)$ .

$\log (5) + \frac{1}{2} \log (2^{2} \cdot 3)$

Step 5

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

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

Step 6

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

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

Step 7

Apply the distributive property.

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

Step 8

Cancel the common factor of $2$ .

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Step 8.1

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

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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

Step 9

Combine $\frac{1}{2}$ and $\log (3)$ .

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