Write as a Single Logarithm 2 natural log of x+ natural log of x+1

$2 \ln (x) + \ln (x + 1)$

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

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

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

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

Apply the distributive property.

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

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $2$ and $1$ .

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

Multiply $x^{2}$ by $1$ .

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