Expand the Logarithmic Expression log base x of ( cube root of x+3)/( square root of x+4)

$\log_{x} (\frac{\sqrt[3]{x + 3}}{\sqrt{x + 4}})$

Rewrite $\log_{x} (\frac{\sqrt[3]{x + 3}}{\sqrt{x + 4}})$ as $\log_{x} (\sqrt[3]{x + 3}) - \log_{x} (\sqrt{x + 4})$ .

$\log_{x} (\sqrt[3]{x + 3}) - \log_{x} (\sqrt{x + 4})$

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

$\log_{x} ({(x + 3)}^{\frac{1}{3}}) - \log_{x} (\sqrt{x + 4})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

$\log_{x} ({(x + 3)}^{\frac{1}{3}}) - \log_{x} ({(x + 4)}^{\frac{1}{2}})$

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

$\frac{1}{3} \log_{x} (x + 3) - \log_{x} ({(x + 4)}^{\frac{1}{2}})$

Expand $\log_{x} ({(x + 4)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{3} \log_{x} (x + 3) - (\frac{1}{2} \log_{x} (x + 4))$

Combine $\frac{1}{2}$ and $\log_{x} (x + 4)$ .

$\frac{1}{3} \log_{x} (x + 3) - \frac{\log_{x} (x + 4)}{2}$

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

$\frac{\log_{x} (x + 3)}{3} - \frac{\log_{x} (x + 4)}{2}$

To write $\frac{\log_{x} (x + 3)}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\log_{x} (x + 3)}{3} \cdot \frac{2}{2} - \frac{\log_{x} (x + 4)}{2}$

To write $- \frac{\log_{x} (x + 4)}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\log_{x} (x + 3)}{3} \cdot \frac{2}{2} - \frac{\log_{x} (x + 4)}{2} \cdot \frac{3}{3}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{\log_{x} (x + 3)}{3}$ by $\frac{2}{2}$ .

$\frac{\log_{x} (x + 3) \cdot 2}{3 \cdot 2} - \frac{\log_{x} (x + 4)}{2} \cdot \frac{3}{3}$

Multiply $3$ by $2$ .

$\frac{\log_{x} (x + 3) \cdot 2}{6} - \frac{\log_{x} (x + 4)}{2} \cdot \frac{3}{3}$

Multiply $\frac{\log_{x} (x + 4)}{2}$ by $\frac{3}{3}$ .

$\frac{\log_{x} (x + 3) \cdot 2}{6} - \frac{\log_{x} (x + 4) \cdot 3}{2 \cdot 3}$

Multiply $2$ by $3$ .

$\frac{\log_{x} (x + 3) \cdot 2}{6} - \frac{\log_{x} (x + 4) \cdot 3}{6}$

Combine the numerators over the common denominator.

$\frac{\log_{x} (x + 3) \cdot 2 - \log_{x} (x + 4) \cdot 3}{6}$

Simplify the numerator.

Move $2$ to the left of $\log_{x} (x + 3)$ .

$\frac{2 \cdot \log_{x} (x + 3) - \log_{x} (x + 4) \cdot 3}{6}$

Multiply $3$ by $- 1$ .

$\frac{2 \log_{x} (x + 3) - 3 \log_{x} (x + 4)}{6}$