Expand the Logarithmic Expression log of square root of (yz^3)/(x^5)

$\log (\sqrt{\frac{y z^{3}}{x^{5}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{y z^{3}}{x^{5}}}$ as ${(\frac{y z^{3}}{x^{5}})}^{\frac{1}{2}}$ .

$\log ({(\frac{y z^{3}}{x^{5}})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(\frac{y z^{3}}{x^{5}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (\frac{y z^{3}}{x^{5}})$

Step 3

Rewrite $\log (\frac{y z^{3}}{x^{5}})$ as $\log (y z^{3}) - \log (x^{5})$ .

$\frac{1}{2} (\log (y z^{3}) - \log (x^{5}))$

Step 4

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

$\frac{1}{2} (\log (y z^{3}) - (5 \log (x)))$

Step 5

Multiply $5$ by $- 1$ .

$\frac{1}{2} (\log (y z^{3}) - 5 \log (x))$

Step 6

Rewrite $\log (y z^{3})$ as $\log (y) + \log (z^{3})$ .

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

Step 7

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

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

Step 8

Apply the distributive property.

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

Step 9

Simplify.

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

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

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

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

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

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

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

Multiply $\frac{1}{2} (- 5 \log (x))$ .

Combine $- 5$ and $\frac{1}{2}$ .

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

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

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

Step 10

Move the negative in front of the fraction.

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