Simplify/Condense 1/3(2 natural log of x+3+ natural log of x- natural log of x^2-1)

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

Step 1

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

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

Step 2

Simplify each term.

Simplify $2 \ln (x + 3)$ by moving $2$ inside the logarithm.

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

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify $\frac{1}{3} \ln (\frac{{(x + 3)}^{2} x}{(x + 1) (x - 1)})$ by moving $\frac{1}{3}$ inside the logarithm.

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

Step 6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{{(x + 3)}^{2} x}{(x + 1) (x - 1)}$ .

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

Apply the product rule to ${(x + 3)}^{2} x$ .

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

Apply the product rule to $(x + 1) (x - 1)$ .

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

Step 7

Multiply the exponents in ${({(x + 3)}^{2})}^{\frac{1}{3}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $2$ and $\frac{1}{3}$ .

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