Precalculus Examples

$\frac{1}{3} \cdot \log ({(x + 2)}^{3}) + \frac{1}{2} \cdot (\log (x^{4}) - \log ({(x^{2} - x - 6)}^{2}))$

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.

$\log ({({(x + 2)}^{3})}^{\frac{1}{3}}) + \frac{1}{2} \cdot (\log (x^{4}) - \log ({(x^{2} - x - 6)}^{2}))$

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, ${(a^{m})}^{n} = a^{m n}$ .

$\log ({(x + 2)}^{3 (\frac{1}{3})}) + \frac{1}{2} \cdot (\log (x^{4}) - \log ({(x^{2} - x - 6)}^{2}))$

Step 1.2.2

Cancel the common factor of $3$ .

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Step 1.2.2.1

Cancel the common factor.

$\log ({(x + 2)}^{3 (\frac{1}{3})}) + \frac{1}{2} \cdot (\log (x^{4}) - \log ({(x^{2} - x - 6)}^{2}))$

Step 1.2.2.2

Rewrite the expression.

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

Step 1.3

Simplify.

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

Step 1.4

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

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

Step 1.5

Simplify the denominator.

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Step 1.5.1

Factor $x^{2} - x - 6$ using the AC method.

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Step 1.5.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 1.5.1.2

Write the factored form using these integers.

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

Step 1.5.2

Apply the product rule to $(x - 3) (x + 2)$ .

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

Step 1.6

Simplify $\frac{1}{2} \log (\frac{x^{4}}{{(x - 3)}^{2} {(x + 2)}^{2}})$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.7.1

Apply the product rule to $\frac{x^{4}}{{(x - 3)}^{2} {(x + 2)}^{2}}$ .

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

Step 1.7.2

Apply the product rule to ${(x - 3)}^{2} {(x + 2)}^{2}$ .

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

Step 1.8

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

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Step 1.8.1

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

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

Step 1.8.2

Cancel the common factor of $2$ .

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Step 1.8.2.1

Factor $2$ out of $4$ .

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

Step 1.8.2.2

Cancel the common factor.

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

Step 1.8.2.3

Rewrite the expression.

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

Step 1.9

Simplify the denominator.

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Step 1.9.1

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

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Step 1.9.1.1

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

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

Step 1.9.1.2

Cancel the common factor of $2$ .

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Step 1.9.1.2.1

Cancel the common factor.

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

Step 1.9.1.2.2

Rewrite the expression.

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

Step 1.9.2

Simplify.

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

Step 1.9.3

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

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Step 1.9.3.1

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

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

Step 1.9.3.2

Cancel the common factor of $2$ .

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Step 1.9.3.2.1

Cancel the common factor.

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

Step 1.9.3.2.2

Rewrite the expression.

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

Step 1.9.4

Simplify.

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

Step 2

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

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

Step 3

Cancel the common factor of $x + 2$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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