Write the expression as the logarithm of a single number or expression. Assime that all variables represent positive numbers.
\[
3 \ln (x+6)-\ln (x+2)
\]
\[
3 \ln (x+6)-\ln (x+2)=
\]
(Simplify your answer.)

Step 1 ：First, we can use the properties of logarithms to simplify the expression. The property \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\) allows us to combine the two terms into one logarithm.

Step 2 ：So, \(3 \ln (x+6)-\ln (x+2) = \ln\left((x+6)^3\right) - \ln(x+2)\).

Step 3 ：Then, we can use the same property again to combine these two logarithms into one: \(\ln\left((x+6)^3\right) - \ln(x+2) = \ln\left(\frac{(x+6)^3}{x+2}\right)\).

Step 4 ：So, the expression \(3 \ln (x+6)-\ln (x+2)\) simplifies to \(\ln\left(\frac{(x+6)^3}{x+2}\right)\).

Step 5 ：Finally, we check our result. The original expression and the final expression are equivalent, so our solution is correct. The final answer is \(\boxed{\ln\left(\frac{(x+6)^3}{x+2}\right)}\).