Simplify/Condense 3 log base 5 of x-1/2 log base 5 of 6-x

$3 \log_{5} (x) - \frac{1}{2} \log_{5} (6 - x)$

Step 1

Simplify each term.

Simplify $3 \log_{5} (x)$ by moving $3$ inside the logarithm.

$\log_{5} (x^{3}) - \frac{1}{2} \cdot \log_{5} (6 - x)$

Multiply $- \frac{1}{2} \log_{5} (6 - x)$ .

Reorder $\log_{5} (6 - x)$ and $\frac{1}{2}$ .

$\log_{5} (x^{3}) - (\frac{1}{2} \log_{5} (6 - x))$

Simplify $\frac{1}{2} \log_{5} (6 - x)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{5} (x^{3}) - \log_{5} ({(6 - x)}^{\frac{1}{2}})$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{x^{3}}{{(6 - x)}^{\frac{1}{2}}})$