Precalculus Examples

$9 \log_{2} (\sqrt[3]{x^{2} - 4}) - 2 \log_{2} (x - 2) + \log_{2} (x)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite $4$ as $2^{2}$ .

$9 \log_{2} (\sqrt[3]{x^{2} - 2^{2}}) - 2 \log_{2} (x - 2) + \log_{2} (x)$

Step 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 $b = 2$ .

$9 \log_{2} (\sqrt[3]{(x + 2) (x - 2)}) - 2 \log_{2} (x - 2) + \log_{2} (x)$

Step 1.3

Simplify $9 \log_{2} (\sqrt[3]{(x + 2) (x - 2)})$ by moving $9$ inside the logarithm.

$\log_{2} ({\sqrt[3]{(x + 2) (x - 2)}}^{9}) - 2 \log_{2} (x - 2) + \log_{2} (x)$

Step 1.4

Rewrite ${\sqrt[3]{(x + 2) (x - 2)}}^{9}$ as ${((x + 2) (x - 2))}^{3}$ .

Tap for more steps...

Step 1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{(x + 2) (x - 2)}$ as ${((x + 2) (x - 2))}^{\frac{1}{3}}$ .

$\log_{2} ({({((x + 2) (x - 2))}^{\frac{1}{3}})}^{9}) - 2 \log_{2} (x - 2) + \log_{2} (x)$

Step 1.4.2

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

$\log_{2} ({((x + 2) (x - 2))}^{\frac{1}{3} \cdot 9}) - 2 \log_{2} (x - 2) + \log_{2} (x)$

Step 1.4.3

Combine $\frac{1}{3}$ and $9$ .

$\log_{2} ({((x + 2) (x - 2))}^{\frac{9}{3}}) - 2 \log_{2} (x - 2) + \log_{2} (x)$

Step 1.4.4

Cancel the common factor of $9$ and $3$ .

Tap for more steps...

Step 1.4.4.1

Factor $3$ out of $9$ .

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

Step 1.4.4.2

Cancel the common factors.

Tap for more steps...

Step 1.4.4.2.1

Factor $3$ out of $3$ .

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

Step 1.4.4.2.2

Cancel the common factor.

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

Step 1.4.4.2.3

Rewrite the expression.

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

Step 1.4.4.2.4

Divide $3$ by $1$ .

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

Step 1.5

Expand $(x + 2) (x - 2)$ using the FOIL Method.

Tap for more steps...

Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

Step 1.6

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

Tap for more steps...

Step 1.6.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

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

Step 1.6.2

Add $- 2 x$ and $2 x$ .

$\log_{2} ({(x \cdot x + 0 + 2 \cdot - 2)}^{3}) - 2 \log_{2} (x - 2) + \log_{2} (x)$

Step 1.6.3

Add $x \cdot x$ and $0$ .

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

Step 1.7

Simplify each term.

Tap for more steps...

Step 1.7.1

Multiply $x$ by $x$ .

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

Step 1.7.2

Multiply $2$ by $- 2$ .

$\log_{2} ({(x^{2} - 4)}^{3}) - 2 \log_{2} (x - 2) + \log_{2} (x)$

Step 1.8

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

$\log_{2} ({(x^{2} - 4)}^{3}) - \log_{2} ({(x - 2)}^{2}) + \log_{2} (x)$

Step 2

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

$\log_{2} (\frac{{(x^{2} - 4)}^{3}}{{(x - 2)}^{2}}) + \log_{2} (x)$

Step 3

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

$\log_{2} (\frac{{(x^{2} - 4)}^{3}}{{(x - 2)}^{2}} x)$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

Step 4.3

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

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

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Cancel the common factor of ${(x - 2)}^{3}$ and ${(x - 2)}^{2}$ .

Tap for more steps...

Step 5.1.1

Factor ${(x - 2)}^{2}$ out of ${(x + 2)}^{3} {(x - 2)}^{3}$ .

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

Step 5.1.2

Cancel the common factors.

Tap for more steps...

Step 5.1.2.1

Multiply by $1$ .

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

Step 5.1.2.2

Cancel the common factor.

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

Step 5.1.2.3

Rewrite the expression.

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

Step 5.1.2.4

Divide ${(x + 2)}^{3} (x - 2)$ by $1$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Move $- 2$ to the left of ${(x + 2)}^{3}$ .

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

Step 5.4

Apply the distributive property.

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

Step 6

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

Tap for more steps...

Step 6.1

Move $x$ .

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 7

Reorder factors in $\log_{2} ({(x + 2)}^{3} x^{2} - 2 {(x + 2)}^{3} x)$ .

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