Precalculus Examples

$3 \log (2 x) + \log (x^{2} - 1) - 2 \log (x + 1) - 2 \log (8)$

Step 1

Simplify each term.

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Step 1.1

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

$\log ({(2 x)}^{3}) + \log (x^{2} - 1) - 2 \log (x + 1) - 2 \log (8)$

Step 1.2

Apply the product rule to $2 x$ .

$\log (2^{3} x^{3}) + \log (x^{2} - 1) - 2 \log (x + 1) - 2 \log (8)$

Step 1.3

Raise $2$ to the power of $3$ .

$\log (8 x^{3}) + \log (x^{2} - 1) - 2 \log (x + 1) - 2 \log (8)$

Step 1.4

Simplify $- 2 \log (x + 1)$ by moving $2$ inside the logarithm.

$\log (8 x^{3}) + \log (x^{2} - 1) - \log ({(x + 1)}^{2}) - 2 \log (8)$

Step 1.5

Simplify $- 2 \log (8)$ by moving $2$ inside the logarithm.

$\log (8 x^{3}) + \log (x^{2} - 1) - \log ({(x + 1)}^{2}) - \log (8^{2})$

Step 1.6

Raise $8$ to the power of $2$ .

$\log (8 x^{3}) + \log (x^{2} - 1) - \log ({(x + 1)}^{2}) - \log (64)$

Step 2

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

$\log (8 x^{3} (x^{2} - 1)) - \log ({(x + 1)}^{2}) - \log (64)$

Step 3

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

$\log (\frac{8 x^{3} (x^{2} - 1)}{{(x + 1)}^{2}}) - \log (64)$

Step 4

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

$\log (\frac{\frac{8 x^{3} (x^{2} - 1)}{{(x + 1)}^{2}}}{64})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{8 x^{3} (x^{2} - 1)}{{(x + 1)}^{2}} \cdot \frac{1}{64})$

Step 6

Simplify terms.

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Step 6.1

Combine.

$\log (\frac{8 x^{3} (x^{2} - 1) \cdot 1}{{(x + 1)}^{2} \cdot 64})$

Step 6.2

Cancel the common factor of $8$ and $64$ .

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Step 6.2.1

Factor $8$ out of $8 x^{3} (x^{2} - 1) \cdot 1$ .

$\log (\frac{8 ((x^{3} (x^{2} - 1)) \cdot 1)}{{(x + 1)}^{2} \cdot 64})$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $8$ out of ${(x + 1)}^{2} \cdot 64$ .

$\log (\frac{8 ((x^{3} (x^{2} - 1)) \cdot 1)}{8 ({(x + 1)}^{2} \cdot 8)})$

Step 6.2.2.2

Cancel the common factor.

$\log (\frac{8 ((x^{3} (x^{2} - 1)) \cdot 1)}{8 ({(x + 1)}^{2} \cdot 8)})$

Step 6.2.2.3

Rewrite the expression.

$\log (\frac{(x^{3} (x^{2} - 1)) \cdot 1}{{(x + 1)}^{2} \cdot 8})$

Step 6.3

Multiply $x^{3}$ by $1$ .

$\log (\frac{x^{3} (x^{2} - 1)}{{(x + 1)}^{2} \cdot 8})$

Step 7

Simplify the numerator.

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Step 7.1

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

$\log (\frac{x^{3} (x^{2} - 1^{2})}{{(x + 1)}^{2} \cdot 8})$

Step 7.2

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 $b = 1$ .

$\log (\frac{x^{3} (x + 1) (x - 1)}{{(x + 1)}^{2} \cdot 8})$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Cancel the common factor of $x + 1$ and ${(x + 1)}^{2}$ .

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Step 8.1.1

Factor $x + 1$ out of $x^{3} (x + 1) (x - 1)$ .

$\log (\frac{(x + 1) (x^{3} (x - 1))}{{(x + 1)}^{2} \cdot 8})$

Step 8.1.2

Cancel the common factors.

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Step 8.1.2.1

Factor $x + 1$ out of ${(x + 1)}^{2} \cdot 8$ .

$\log (\frac{(x + 1) (x^{3} (x - 1))}{(x + 1) ((x + 1) \cdot 8)})$

Step 8.1.2.2

Cancel the common factor.

$\log (\frac{(x + 1) (x^{3} (x - 1))}{(x + 1) ((x + 1) \cdot 8)})$

Step 8.1.2.3

Rewrite the expression.

$\log (\frac{x^{3} (x - 1)}{(x + 1) \cdot 8})$

Step 8.2

Move $8$ to the left of $x + 1$ .

$\log (\frac{x^{3} (x - 1)}{8 (x + 1)})$