Expand the Logarithmic Expression log base 3 of ( square root of 3x+2y^2)/(5z^2)

$\log_{3} (\frac{\sqrt{3 x + 2 y^{2}}}{5 z^{2}})$

Step 1

Rewrite $\log_{3} (\frac{\sqrt{3 x + 2 y^{2}}}{5 z^{2}})$ as $\log_{3} (\sqrt{3 x + 2 y^{2}}) - \log_{3} (5 z^{2})$ .

$\log_{3} (\sqrt{3 x + 2 y^{2}}) - \log_{3} (5 z^{2})$

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\log_{3} (5 z^{2})$ as $\log_{3} (5) + \log_{3} (z^{2})$ .

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

Step 5

Expand $\log_{3} (z^{2})$ by moving $2$ outside the logarithm.

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

Step 6

Simplify each term.

Combine $\frac{1}{2}$ and $\log_{3} (3 x + 2 y^{2})$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

Step 7

To write $- \log_{3} (5)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8

Combine $- \log_{3} (5)$ and $\frac{2}{2}$ .

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Multiply $2$ by $- 1$ .

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