Precalculus Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $2 \log_{3} (2 x - 2)$ by moving $2$ inside the logarithm.

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

Step 1.2

Simplify $- 3 \log_{3} (x - 1)$ by moving $3$ inside the logarithm.

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

Step 2

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

$\log_{3} ((x^{2} - 1) {(2 x - 2)}^{2}) - \log_{3} ({(x - 1)}^{3})$

Step 3

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

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

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

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

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

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 $b = 1$ .

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

Step 4.3

Factor $2$ out of $2 x - 2$ .

Tap for more steps...

Step 4.3.1

Factor $2$ out of $2 x$ .

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

Step 4.3.2

Factor $2$ out of $- 2$ .

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

Step 4.3.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

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

Step 4.4

Apply the product rule to $2 (x - 1)$ .

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

Step 4.5

Multiply $x - 1$ by ${(x - 1)}^{2}$ by adding the exponents.

Tap for more steps...

Step 4.5.1

Move ${(x - 1)}^{2}$ .

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

Step 4.5.2

Multiply ${(x - 1)}^{2}$ by $x - 1$ .

Tap for more steps...

Step 4.5.2.1

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

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

Step 4.5.2.2

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

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

Step 4.5.3

Add $2$ and $1$ .

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

Step 4.6

Raise $2$ to the power of $2$ .

$\log_{3} (\frac{(x + 1) {(x - 1)}^{3} \cdot 4}{{(x - 1)}^{3}})$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Cancel the common factor of ${(x - 1)}^{3}$ .

Tap for more steps...

Step 5.1.1

Cancel the common factor.

$\log_{3} (\frac{(x + 1) {(x - 1)}^{3} \cdot 4}{{(x - 1)}^{3}})$

Step 5.1.2

Divide $(x + 1) \cdot 4$ by $1$ .

$\log_{3} ((x + 1) \cdot 4)$

Step 5.2

Apply the distributive property.

$\log_{3} (x \cdot 4 + 1 \cdot 4)$

Step 5.3

Simplify the expression.

Tap for more steps...

Step 5.3.1

Move $4$ to the left of $x$ .

$\log_{3} (4 \cdot x + 1 \cdot 4)$

Step 5.3.2

Multiply $4$ by $1$ .

$\log_{3} (4 x + 4)$