Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\log (x y z)$ as $\log (x y) + \log (z)$ .

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

Step 5

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

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

Step 6

Simplify each term.

Tap for more steps...

Step 6.1

Apply the distributive property.

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

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify each term.

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Factor $\log (z)$ out of $\log (z) - \log (z) \cdot 2$ .

Tap for more steps...

Step 10.1.1.1

Multiply by $1$ .

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

Step 10.1.1.2

Factor $\log (z)$ out of $- \log (z) \cdot 2$ .

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

Step 10.1.1.3

Factor $\log (z)$ out of $\log (z) \cdot 1 + \log (z) (- 1 \cdot 2)$ .

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

Step 10.1.2

Multiply $- 1$ by $2$ .

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

Step 10.1.3

Subtract $2$ from $1$ .

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

Step 10.2

Move $- 1$ to the left of $\log (z)$ .

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

Step 10.3

Move the negative in front of the fraction.

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