Precalculus Examples

$\log (x) + \log (7) + \log (x^{2} - 1) - \log (x + 1)$

Step 1

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

$\log (x \cdot 7) + \log (x^{2} - 1) - \log (x + 1)$

Step 2

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

$\log (x \cdot 7 (x^{2} - 1)) - \log (x + 1)$

Step 3

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

$\log (\frac{x \cdot 7 (x^{2} - 1)}{x + 1})$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

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

$\log (\frac{x \cdot 7 (x^{2} - 1^{2})}{x + 1})$

Step 4.2

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 $b = 1$ .

$\log (\frac{x \cdot 7 (x + 1) (x - 1)}{x + 1})$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 5.1.1

Cancel the common factor.

$\log (\frac{x \cdot 7 (x + 1) (x - 1)}{x + 1})$

Step 5.1.2

Divide $x \cdot 7 (x - 1)$ by $1$ .

$\log (x \cdot 7 (x - 1))$

Step 5.2

Move $7$ to the left of $x$ .

$\log (7 \cdot x (x - 1))$

Step 5.3

Apply the distributive property.

$\log (7 x \cdot x + 7 x \cdot - 1)$

Step 6

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 6.1

Move $x$ .

$\log (7 (x \cdot x) + 7 x \cdot - 1)$

Step 6.2

Multiply $x$ by $x$ .

$\log (7 x^{2} + 7 x \cdot - 1)$

Step 7

Multiply $- 1$ by $7$ .

$\log (7 x^{2} - 7 x)$