Simplify/Condense 3 natural log of x+1-2 natural log of y+ natural log of y+ natural log of 2

$3 \ln (x + 1) - 2 \ln (y) + \ln (y) + \ln (2)$

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

$3 \ln (x + 1) - 2 \ln (y) + \ln (y \cdot 2)$

Simplify each term.

Simplify $3 \ln (x + 1)$ by moving $3$ inside the logarithm.

$\ln ({(x + 1)}^{3}) - 2 \ln (y) + \ln (y \cdot 2)$

Simplify $- 2 \ln (y)$ by moving $2$ inside the logarithm.

$\ln ({(x + 1)}^{3}) - \ln (y^{2}) + \ln (y \cdot 2)$

Move $2$ to the left of $y$ .

$\ln ({(x + 1)}^{3}) - \ln (y^{2}) + \ln (2 y)$

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

$\ln (\frac{{(x + 1)}^{3}}{y^{2}}) + \ln (2 y)$

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

$\ln (\frac{{(x + 1)}^{3}}{y^{2}} (2 y))$

Rewrite using the commutative property of multiplication.

$\ln (2 \frac{{(x + 1)}^{3}}{y^{2}} y)$

Combine $2$ and $\frac{{(x + 1)}^{3}}{y^{2}}$ .

$\ln (\frac{2 {(x + 1)}^{3}}{y^{2}} y)$

Cancel the common factor of $y$ .

Factor $y$ out of $y^{2}$ .

$\ln (\frac{2 {(x + 1)}^{3}}{y \cdot y} y)$

Cancel the common factor.

$\ln (\frac{2 {(x + 1)}^{3}}{y \cdot y} y)$

Rewrite the expression.

$\ln (\frac{2 {(x + 1)}^{3}}{y})$