Expand the Logarithmic Expression log of ( cube root of x)/( cube root of yz)

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

Step 1

Rewrite $\log (\frac{\sqrt[3]{x}}{\sqrt[3]{y z}})$ as $\log (\sqrt[3]{x}) - \log (\sqrt[3]{y z})$ .

$\log (\sqrt[3]{x}) - \log (\sqrt[3]{y z})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\log (x^{\frac{1}{3}}) - \log (\sqrt[3]{y z})$

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Apply the distributive property.

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