Expand the Logarithmic Expression log base 5 of (9 square root of x)/(y^3)

$\log_{5} (\frac{9 \sqrt{x}}{y^{3}})$

Step 1

Rewrite $\log_{5} (\frac{9 \sqrt{x}}{y^{3}})$ as $\log_{5} (9 \sqrt{x}) - \log_{5} (y^{3})$ .

$\log_{5} (9 \sqrt{x}) - \log_{5} (y^{3})$

Step 2

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

$\log_{5} (9 x^{\frac{1}{2}}) - \log_{5} (y^{3})$

Step 3

Expand $\log_{5} (y^{3})$ by moving $3$ outside the logarithm.

$\log_{5} (9 x^{\frac{1}{2}}) - (3 \log_{5} (y))$

Step 4

Multiply $3$ by $- 1$ .

$\log_{5} (9 x^{\frac{1}{2}}) - 3 \log_{5} (y)$

Step 5

Rewrite $\log_{5} (9 x^{\frac{1}{2}})$ as $\log_{5} (9) + \log_{5} (x^{\frac{1}{2}})$ .

$\log_{5} (9) + \log_{5} (x^{\frac{1}{2}}) - 3 \log_{5} (y)$

Step 6

Expand $\log_{5} (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{5} (9) + \frac{1}{2} \log_{5} (x) - 3 \log_{5} (y)$

Step 7

Simplify each term.

Rewrite $\log_{5} (9)$ as $\log_{5} (3^{2})$ .

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

Expand $\log_{5} (3^{2})$ by moving $2$ outside the logarithm.

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

Combine $\frac{1}{2}$ and $\log_{5} (x)$ .

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