Precalculus Examples

$2 \log_{2} (x) + \frac{1}{3} \cdot (\log (2) (27 x - 27)) - 2 \log_{2} (x^{3} - x)$

Step 1

Simplify each term.

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Step 1.1

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

$\log_{2} (x^{2}) + \frac{1}{3} \cdot (\log (2) (27 x - 27)) - 2 \log_{2} (x^{3} - x)$

Step 1.2

Apply the distributive property.

$\log_{2} (x^{2}) + \frac{1}{3} \cdot (\log (2) (27 x) + \log (2) \cdot - 27) - 2 \log_{2} (x^{3} - x)$

Step 1.3

Simplify $27 \log (2)$ by moving $27$ inside the logarithm.

$\log_{2} (x^{2}) + \frac{1}{3} \cdot (\log (2^{27}) x + \log (2) \cdot - 27) - 2 \log_{2} (x^{3} - x)$

Step 1.4

Multiply $\log (2) \cdot - 27$ .

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Step 1.4.1

Reorder $\log (2)$ and $- 27$ .

$\log_{2} (x^{2}) + \frac{1}{3} \cdot (\log (2^{27}) x - 27 \cdot \log (2)) - 2 \log_{2} (x^{3} - x)$

Step 1.4.2

Simplify $- 27 \cdot \log (2)$ by moving $27$ inside the logarithm.

$\log_{2} (x^{2}) + \frac{1}{3} \cdot (\log (2^{27}) x - \log (2^{27})) - 2 \log_{2} (x^{3} - x)$

Step 1.5

Simplify each term.

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Step 1.5.1

Raise $2$ to the power of $27$ .

$\log_{2} (x^{2}) + \frac{1}{3} \cdot (\log (134217728) x - \log (2^{27})) - 2 \log_{2} (x^{3} - x)$

Step 1.5.2

Raise $2$ to the power of $27$ .

$\log_{2} (x^{2}) + \frac{1}{3} \cdot (\log (134217728) x - \log (134217728)) - 2 \log_{2} (x^{3} - x)$

Step 1.6

Apply the distributive property.

$\log_{2} (x^{2}) + \frac{1}{3} (\log (134217728) x) + \frac{1}{3} (- \log (134217728)) - 2 \log_{2} (x^{3} - x)$

Step 1.7

Multiply $\frac{1}{3} (\log (134217728) x)$ .

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Step 1.7.1

Reorder $\log (134217728)$ and $\frac{1}{3}$ .

$\log_{2} (x^{2}) + \frac{1}{3} \log (134217728) x + \frac{1}{3} (- \log (134217728)) - 2 \log_{2} (x^{3} - x)$

Step 1.7.2

Simplify $\frac{1}{3} \log (134217728)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{2} (x^{2}) + \log (134217728^{\frac{1}{3}}) x + \frac{1}{3} (- \log (134217728)) - 2 \log_{2} (x^{3} - x)$

Step 1.8

Simplify $\frac{1}{3} \log (134217728)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{2} (x^{2}) + \log (134217728^{\frac{1}{3}}) x - \log (134217728^{\frac{1}{3}}) - 2 \log_{2} (x^{3} - x)$

Step 1.9

Simplify each term.

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Step 1.9.1

Rewrite $134217728$ as $512^{3}$ .

$\log_{2} (x^{2}) + \log ({(512^{3})}^{\frac{1}{3}}) x - \log (134217728^{\frac{1}{3}}) - 2 \log_{2} (x^{3} - x)$

Step 1.9.2

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

$\log_{2} (x^{2}) + \log (512^{3 (\frac{1}{3})}) x - \log (134217728^{\frac{1}{3}}) - 2 \log_{2} (x^{3} - x)$

Step 1.9.3

Cancel the common factor of $3$ .

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Step 1.9.3.1

Cancel the common factor.

$\log_{2} (x^{2}) + \log (512^{3 (\frac{1}{3})}) x - \log (134217728^{\frac{1}{3}}) - 2 \log_{2} (x^{3} - x)$

Step 1.9.3.2

Rewrite the expression.

$\log_{2} (x^{2}) + \log (512^{1}) x - \log (134217728^{\frac{1}{3}}) - 2 \log_{2} (x^{3} - x)$

Step 1.9.4

Evaluate the exponent.

$\log_{2} (x^{2}) + \log (512) x - \log (134217728^{\frac{1}{3}}) - 2 \log_{2} (x^{3} - x)$

Step 1.9.5

Rewrite $134217728$ as $512^{3}$ .

$\log_{2} (x^{2}) + \log (512) x - \log ({(512^{3})}^{\frac{1}{3}}) - 2 \log_{2} (x^{3} - x)$

Step 1.9.6

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

$\log_{2} (x^{2}) + \log (512) x - \log (512^{3 (\frac{1}{3})}) - 2 \log_{2} (x^{3} - x)$

Step 1.9.7

Cancel the common factor of $3$ .

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Step 1.9.7.1

Cancel the common factor.

$\log_{2} (x^{2}) + \log (512) x - \log (512^{3 (\frac{1}{3})}) - 2 \log_{2} (x^{3} - x)$

Step 1.9.7.2

Rewrite the expression.

$\log_{2} (x^{2}) + \log (512) x - \log (512^{1}) - 2 \log_{2} (x^{3} - x)$

Step 1.9.8

Evaluate the exponent.

$\log_{2} (x^{2}) + \log (512) x - \log (512) - 2 \log_{2} (x^{3} - x)$

Step 1.10

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

$\log_{2} (x^{2}) + \log (512) x - \log (512) - \log_{2} ({(x^{3} - x)}^{2})$

Step 2

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

$\log_{2} (\frac{x^{2}}{{(x^{3} - x)}^{2}}) + \log (512) x - \log (512)$

Step 3

Simplify each term.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

Factor $x$ out of $x^{3} - x$ .

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Step 3.1.1.1

Factor $x$ out of $x^{3}$ .

$\log_{2} (\frac{x^{2}}{{(x \cdot x^{2} - x)}^{2}}) + \log (512) x - \log (512)$

Step 3.1.1.2

Factor $x$ out of $- x$ .

$\log_{2} (\frac{x^{2}}{{(x \cdot x^{2} + x \cdot - 1)}^{2}}) + \log (512) x - \log (512)$

Step 3.1.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 1$ .

$\log_{2} (\frac{x^{2}}{{(x (x^{2} - 1))}^{2}}) + \log (512) x - \log (512)$

Step 3.1.2

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

$\log_{2} (\frac{x^{2}}{{(x (x^{2} - 1^{2}))}^{2}}) + \log (512) x - \log (512)$

Step 3.1.3

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 $b = 1$ .

$\log_{2} (\frac{x^{2}}{{(x (x + 1) (x - 1))}^{2}}) + \log (512) x - \log (512)$

Step 3.1.4

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

$\log_{2} (\frac{x^{2}}{{(x (x + 1))}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$

Step 3.1.5

Apply the distributive property.

$\log_{2} (\frac{x^{2}}{{(x \cdot x + x \cdot 1)}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$

Step 3.1.6

Multiply $x$ by $x$ .

$\log_{2} (\frac{x^{2}}{{(x^{2} + x \cdot 1)}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$

Step 3.1.7

Multiply $x$ by $1$ .

$\log_{2} (\frac{x^{2}}{{(x^{2} + x)}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$

Step 3.1.8

Factor $x$ out of $x^{2} + x$ .

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Step 3.1.8.1

Factor $x$ out of $x^{2}$ .

$\log_{2} (\frac{x^{2}}{{(x \cdot x + x)}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$

Step 3.1.8.2

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

$\log_{2} (\frac{x^{2}}{{(x \cdot x + x^{1})}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$

Step 3.1.8.3

Factor $x$ out of $x^{1}$ .

$\log_{2} (\frac{x^{2}}{{(x \cdot x + x \cdot 1)}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$

Step 3.1.8.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$\log_{2} (\frac{x^{2}}{{(x (x + 1))}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$

Step 3.1.9

Apply the product rule to $x (x + 1)$ .

$\log_{2} (\frac{x^{2}}{x^{2} {(x + 1)}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$

Step 3.2

Cancel the common factor of $x^{2}$ .

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Step 3.2.1

Cancel the common factor.

$\log_{2} (\frac{x^{2}}{x^{2} {(x + 1)}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$

Step 3.2.2

Rewrite the expression.

$\log_{2} (\frac{1}{{(x + 1)}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$

Step 4

Reorder factors in $\log_{2} (\frac{1}{{(x + 1)}^{2} {(x - 1)}^{2}}) + \log (512) x - \log (512)$ .

$\log_{2} (\frac{1}{{(x + 1)}^{2} {(x - 1)}^{2}}) + x \log (512) - \log (512)$