Expand the Logarithmic Expression log of (x^5)/( square root of zy)

$\log (\frac{x^{5}}{\sqrt{z y}})$

Step 1

Rewrite $\log (\frac{x^{5}}{\sqrt{z y}})$ as $\log (x^{5}) - \log (\sqrt{z y})$ .

$\log (x^{5}) - \log (\sqrt{z y})$

Step 2

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

$\log (x^{5}) - \log ({(z y)}^{\frac{1}{2}})$

Step 3

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

$5 \log (x) - \log ({(z y)}^{\frac{1}{2}})$

Step 4

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

$5 \log (x) - (\frac{1}{2} \log (z y))$

Step 5

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

$5 \log (x) - \frac{\log (z y)}{2}$

Step 6

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

$5 \log (x) - \frac{\log (z) + \log (y)}{2}$

Step 7

To write $5 \log (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 \log (x) \cdot \frac{2}{2} - \frac{\log (z) + \log (y)}{2}$

Step 8

Combine $5 \log (x)$ and $\frac{2}{2}$ .

$\frac{5 \log (x) \cdot 2}{2} - \frac{\log (z) + \log (y)}{2}$

Step 9

Combine the numerators over the common denominator.

$\frac{5 \log (x) \cdot 2 - (\log (z) + \log (y))}{2}$

Step 10

Simplify the numerator.

Multiply $2$ by $5$ .

$\frac{10 \log (x) - (\log (z) + \log (y))}{2}$

Apply the distributive property.

$\frac{10 \log (x) - \log (z) - \log (y)}{2}$