Write as a Single Logarithm 2 log base a of z+1+ log base a of 3z+2

$2 \log_{a} (z + 1) + \log_{a} (3 z + 2)$

Simplify $2 \log_{a} (z + 1)$ by moving $2$ inside the logarithm.

$\log_{a} ({(z + 1)}^{2}) + \log_{a} (3 z + 2)$

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{a} ({(z + 1)}^{2} (3 z + 2))$

Apply the distributive property.

$\log_{a} ({(z + 1)}^{2} (3 z) + {(z + 1)}^{2} \cdot 2)$

Reorder.

Rewrite using the commutative property of multiplication.

$\log_{a} (3 {(z + 1)}^{2} z + {(z + 1)}^{2} \cdot 2)$

Move $2$ to the left of ${(z + 1)}^{2}$ .

$\log_{a} (3 {(z + 1)}^{2} z + 2 \cdot {(z + 1)}^{2})$

Reorder factors in $\log_{a} (3 {(z + 1)}^{2} z + 2 {(z + 1)}^{2})$ .

$\log_{a} (3 z {(z + 1)}^{2} + 2 {(z + 1)}^{2})$