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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine $\frac{1}{3}$ and $\log (y^{5} z^{2})$ .

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

Step 5

Rewrite $\log (y^{5} z^{2})$ as $\log (y^{5}) + \log (z^{2})$ .

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

Step 6

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

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

Step 7

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

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

Step 8

To write $\log (x)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 9

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify the numerator.

Move $3$ to the left of $\log (x)$ .

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

Apply the distributive property.

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

Multiply $5$ by $- 1$ .

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

Multiply $2$ by $- 1$ .

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