Expand the Logarithmic Expression log of cube root of (x^5z)/(y^2)

$\log (\sqrt[3]{\frac{x^{5} z}{y^{2}}})$

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $\log (\frac{x^{5} z}{y^{2}})$ as $\log (x^{5} z) - \log (y^{2})$ .

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

Step 4

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

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

Step 5

Multiply $2$ by $- 1$ .

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

Step 6

Rewrite $\log (x^{5} z)$ as $\log (x^{5}) + \log (z)$ .

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

Step 7

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

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

Step 8

Apply the distributive property.

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

Step 9

Simplify.

Multiply $\frac{1}{3} (5 \log (x))$ .

Combine $5$ and $\frac{1}{3}$ .

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

Combine $\frac{5}{3}$ and $\log (x)$ .

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

Combine $\frac{1}{3}$ and $\log (z)$ .

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

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

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

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

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

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

Step 10

Move the negative in front of the fraction.

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