Write as a Single Logarithm natural log of 2x+2 natural log of x- natural log of 3y

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $x^{2} \frac{2 x}{3 y}$ .

Combine $x^{2}$ and $\frac{2 x}{3 y}$ .

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Step 5

Move $2$ to the left of $x^{3}$ .

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