Precalculus Examples

$\frac{1}{3} \cdot \log ({(x + 2)}^{3}) + \frac{1}{2} \cdot (\log (x^{4}) - \log ({(x^{2 - 2 x - 8})}^{2}))$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{3} \log ({(x + 2)}^{3})$ by moving $\frac{1}{3}$ inside the logarithm.

$\log ({({(x + 2)}^{3})}^{\frac{1}{3}}) + \frac{1}{2} \cdot (\log (x^{4}) - \log ({(x^{2 - 2 x - 8})}^{2}))$

Step 1.2

Multiply the exponents in ${({(x + 2)}^{3})}^{\frac{1}{3}}$ .

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Step 1.2.1

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

$\log ({(x + 2)}^{3 (\frac{1}{3})}) + \frac{1}{2} \cdot (\log (x^{4}) - \log ({(x^{2 - 2 x - 8})}^{2}))$

Step 1.2.2

Cancel the common factor of $3$ .

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Step 1.2.2.1

Cancel the common factor.

$\log ({(x + 2)}^{3 (\frac{1}{3})}) + \frac{1}{2} \cdot (\log (x^{4}) - \log ({(x^{2 - 2 x - 8})}^{2}))$

Step 1.2.2.2

Rewrite the expression.

$\log ({(x + 2)}^{1}) + \frac{1}{2} \cdot (\log (x^{4}) - \log ({(x^{2 - 2 x - 8})}^{2}))$

Step 1.3

Simplify.

$\log (x + 2) + \frac{1}{2} \cdot (\log (x^{4}) - \log ({(x^{2 - 2 x - 8})}^{2}))$

Step 1.4

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

$\log (x + 2) + \frac{1}{2} \cdot \log (\frac{x^{4}}{{(x^{2 - 2 x - 8})}^{2}})$

Step 1.5

Simplify the denominator.

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Step 1.5.1

Subtract $8$ from $2$ .

$\log (x + 2) + \frac{1}{2} \cdot \log (\frac{x^{4}}{{(x^{- 2 x - 6})}^{2}})$

Step 1.5.2

Multiply the exponents in ${(x^{- 2 x - 6})}^{2}$ .

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Step 1.5.2.1

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

$\log (x + 2) + \frac{1}{2} \cdot \log (\frac{x^{4}}{x^{(- 2 x - 6) \cdot 2}})$

Step 1.5.2.2

Apply the distributive property.

$\log (x + 2) + \frac{1}{2} \cdot \log (\frac{x^{4}}{x^{- 2 x \cdot 2 - 6 \cdot 2}})$

Step 1.5.2.3

Multiply $2$ by $- 2$ .

$\log (x + 2) + \frac{1}{2} \cdot \log (\frac{x^{4}}{x^{- 4 x - 6 \cdot 2}})$

Step 1.5.2.4

Multiply $- 6$ by $2$ .

$\log (x + 2) + \frac{1}{2} \cdot \log (\frac{x^{4}}{x^{- 4 x - 12}})$

Step 1.6

Cancel the common factor of $x^{4}$ and $x^{- 4 x - 12}$ .

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Step 1.6.1

Factor $x^{- 4 x - 12}$ out of $x^{4}$ .

$\log (x + 2) + \frac{1}{2} \cdot \log (\frac{x^{- 4 x - 12} x^{4 - (- 4 x - 12)}}{x^{- 4 x - 12}})$

Step 1.6.2

Cancel the common factors.

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Step 1.6.2.1

Multiply by $1$ .

$\log (x + 2) + \frac{1}{2} \cdot \log (\frac{x^{- 4 x - 12} x^{4 - (- 4 x - 12)}}{x^{- 4 x - 12} \cdot 1})$

Step 1.6.2.2

Cancel the common factor.

$\log (x + 2) + \frac{1}{2} \cdot \log (\frac{x^{- 4 x - 12} x^{4 - (- 4 x - 12)}}{x^{- 4 x - 12} \cdot 1})$

Step 1.6.2.3

Rewrite the expression.

$\log (x + 2) + \frac{1}{2} \cdot \log (\frac{x^{4 - (- 4 x - 12)}}{1})$

Step 1.6.2.4

Divide $x^{4 - (- 4 x - 12)}$ by $1$ .

$\log (x + 2) + \frac{1}{2} \cdot \log (x^{4 - (- 4 x - 12)})$

Step 1.7

Simplify each term.

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Step 1.7.1

Apply the distributive property.

$\log (x + 2) + \frac{1}{2} \cdot \log (x^{4 - (- 4 x) - - 12})$

Step 1.7.2

Multiply $- 4$ by $- 1$ .

$\log (x + 2) + \frac{1}{2} \cdot \log (x^{4 + 4 x - - 12})$

Step 1.7.3

Multiply $- 1$ by $- 12$ .

$\log (x + 2) + \frac{1}{2} \cdot \log (x^{4 + 4 x + 12})$

Step 1.8

Add $4$ and $12$ .

$\log (x + 2) + \frac{1}{2} \cdot \log (x^{4 x + 16})$

Step 1.9

Simplify $\frac{1}{2} \log (x^{4 x + 16})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (x + 2) + \log ({(x^{4 x + 16})}^{\frac{1}{2}})$

Step 1.10

Multiply the exponents in ${(x^{4 x + 16})}^{\frac{1}{2}}$ .

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Step 1.10.1

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

$\log (x + 2) + \log (x^{(4 x + 16) \frac{1}{2}})$

Step 1.10.2

Apply the distributive property.

$\log (x + 2) + \log (x^{4 x \frac{1}{2} + 16 (\frac{1}{2})})$

Step 1.10.3

Cancel the common factor of $2$ .

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Step 1.10.3.1

Factor $2$ out of $4 x$ .

$\log (x + 2) + \log (x^{2 (2 x) \frac{1}{2} + 16 (\frac{1}{2})})$

Step 1.10.3.2

Cancel the common factor.

$\log (x + 2) + \log (x^{2 (2 x) \frac{1}{2} + 16 (\frac{1}{2})})$

Step 1.10.3.3

Rewrite the expression.

$\log (x + 2) + \log (x^{2 x + 16 (\frac{1}{2})})$

Step 1.10.4

Cancel the common factor of $2$ .

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Step 1.10.4.1

Factor $2$ out of $16$ .

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

Step 1.10.4.2

Cancel the common factor.

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

Step 1.10.4.3

Rewrite the expression.

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

Step 2

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

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

Step 3

Apply the distributive property.

$\log (x \cdot x^{2 x + 8} + 2 x^{2 x + 8})$

Step 4

Multiply $x$ by $x^{2 x + 8}$ by adding the exponents.

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Step 4.1

Multiply $x$ by $x^{2 x + 8}$ .

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

Add $1$ and $8$ .

$\log (x^{2 x + 9} + 2 x^{2 x + 8})$