Simplify/Condense natural log of a+b+ natural log of a-b-2 natural log of c

$\ln (a + b) + \ln (a - b) - 2 \ln (c)$

Step 1

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

$\ln ((a + b) (a - b)) - 2 \ln (c)$

Step 2

Simplify each term.

Expand $(a + b) (a - b)$ using the FOIL Method.

Apply the distributive property.

$\ln (a (a - b) + b (a - b)) - 2 \ln (c)$

Apply the distributive property.

$\ln (a \cdot a + a (- b) + b (a - b)) - 2 \ln (c)$

Apply the distributive property.

$\ln (a \cdot a + a (- b) + b a + b (- b)) - 2 \ln (c)$

Combine the opposite terms in $a \cdot a + a (- b) + b a + b (- b)$ .

Reorder the factors in the terms $a (- b)$ and $b a$ .

$\ln (a \cdot a - a b + a b + b (- b)) - 2 \ln (c)$

Add $- a b$ and $a b$ .

$\ln (a \cdot a + 0 + b (- b)) - 2 \ln (c)$

Add $a \cdot a$ and $0$ .

$\ln (a \cdot a + b (- b)) - 2 \ln (c)$

Simplify each term.

Multiply $a$ by $a$ .

$\ln (a^{2} + b (- b)) - 2 \ln (c)$

Rewrite using the commutative property of multiplication.

$\ln (a^{2} - b \cdot b) - 2 \ln (c)$

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

$\ln (a^{2} - (b \cdot b)) - 2 \ln (c)$

Multiply $b$ by $b$ .

$\ln (a^{2} - b^{2}) - 2 \ln (c)$

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

$\ln (a^{2} - b^{2}) - \ln (c^{2})$

Step 3

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

$\ln (\frac{a^{2} - b^{2}}{c^{2}})$

Step 4

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

$\ln (\frac{(a + b) (a - b)}{c^{2}})$