Precalculus Examples

$\log (\frac{2 x^{2} z}{\sqrt{y + 2}})$

Step 1

Rewrite $\log (\frac{2 x^{2} z}{\sqrt{y + 2}})$ as $\log (2 x^{2} z) - \log (\sqrt{y + 2})$ .

$\log (2 x^{2} z) - \log (\sqrt{y + 2})$

Step 2

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

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

Step 3

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

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

Step 4

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

$\log (2 x^{2} z) - \frac{\log (y + 2)}{2}$

Step 5

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

$\log (2 x^{2}) + \log (z) - \frac{\log (y + 2)}{2}$

Step 6

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

$\log (2) + \log (x^{2}) + \log (z) - \frac{\log (y + 2)}{2}$

Step 7

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

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

Step 8

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

$2 \log (x) + \log (z) + \log (2) \cdot \frac{2}{2} - \frac{\log (y + 2)}{2}$

Step 9

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

$2 \log (x) + \log (z) + \frac{\log (2) \cdot 2}{2} - \frac{\log (y + 2)}{2}$

Step 10

Combine the numerators over the common denominator.

$2 \log (x) + \log (z) + \frac{\log (2) \cdot 2 - \log (y + 2)}{2}$

Step 11

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

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