Write as a Single Logarithm 4 log base 3 of x-1/3 log base 3 of x+6

$4 \log_{3} (x) - \frac{1}{3} \log_{3} (x + 6)$

Step 1

Simplify each term.

Simplify $4 \log_{3} (x)$ by moving $4$ inside the logarithm.

$\log_{3} (x^{4}) - \frac{1}{3} \cdot \log_{3} (x + 6)$

Multiply $- \frac{1}{3} \log_{3} (x + 6)$ .

Reorder $\log_{3} (x + 6)$ and $\frac{1}{3}$ .

$\log_{3} (x^{4}) - (\frac{1}{3} \log_{3} (x + 6))$

Simplify $\frac{1}{3} \log_{3} (x + 6)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{3} (x^{4}) - \log_{3} ({(x + 6)}^{\frac{1}{3}})$

Step 2

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

$\log_{3} (\frac{x^{4}}{{(x + 6)}^{\frac{1}{3}}})$