Trigonometry Examples

$\log (3 \sqrt{27 y})$

Step 1

Rewrite $\log (3 \sqrt{27 y})$ as $\log (3) + \log (\sqrt{27 y})$ .

$\log (3) + \log (\sqrt{27 y})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{27 y}$ as ${(27 y)}^{\frac{1}{2}}$ .

$\log (3) + \log ({(27 y)}^{\frac{1}{2}})$

Step 3

Expand $\log ({(27 y)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log (3) + \frac{1}{2} \log (27 y)$

Step 4

Rewrite $\log (27 y)$ as $\log (27) + \log (y)$ .

$\log (3) + \frac{1}{2} (\log (27) + \log (y))$

Step 5

Simplify each term.

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

Rewrite $\log (27)$ as $\log (3^{3})$ .

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

Step 5.1.2

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

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

Step 5.2

Apply the distributive property.

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.4

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

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

Step 6

To write $\log (3)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

Tap for more steps...

Step 10.1

Move $2$ to the left of $\log (3)$ .

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

Step 10.2

Add $2 \log (3)$ and $3 \log (3)$ .

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