Precalculus Examples

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

Step 1

Rewrite $\log (\frac{\sqrt{12}}{2^{5}})$ as $\log (\sqrt{12}) - \log (2^{5})$ .

$\log (\sqrt{12}) - \log (2^{5})$

Step 2

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

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

Step 3

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

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

Step 4

Expand $\log (2^{5})$ by moving $5$ outside the logarithm.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Apply the distributive property.

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

Step 9

Cancel the common factor of $2$ .

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Step 9.1

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

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

Step 9.2

Cancel the common factor.

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

Step 9.3

Rewrite the expression.

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

Step 10

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

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

Step 11

Multiply $5$ by $- 1$ .

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

Step 12

Subtract $5 \log (2)$ from $\log (2)$ .

$\frac{\log (3)}{2} - 4 \log (2)$