Precalculus Examples

$\frac{1}{2} \cdot (\log ({(x)}^{4}) - \log ({(x^{2} - 2 x - 15)}^{2}))$

Step 1

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

$\frac{1}{2} \cdot \log (\frac{{(x)}^{4}}{{(x^{2} - 2 x - 15)}^{2}})$

Step 2

Simplify the denominator.

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Step 2.1

Factor $x^{2} - 2 x - 15$ using the AC method.

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Step 2.1.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 2.1.2

Write the factored form using these integers.

$\frac{1}{2} \cdot \log (\frac{x^{4}}{{((x - 5) (x + 3))}^{2}})$

Step 2.2

Apply the product rule to $(x - 5) (x + 3)$ .

$\frac{1}{2} \cdot \log (\frac{x^{4}}{{(x - 5)}^{2} {(x + 3)}^{2}})$

Step 3

Simplify $\frac{1}{2} \log (\frac{x^{4}}{{(x - 5)}^{2} {(x + 3)}^{2}})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log ({(\frac{x^{4}}{{(x - 5)}^{2} {(x + 3)}^{2}})}^{\frac{1}{2}})$

Step 4

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

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Step 4.1

Apply the product rule to $\frac{x^{4}}{{(x - 5)}^{2} {(x + 3)}^{2}}$ .

$\log (\frac{{(x^{4})}^{\frac{1}{2}}}{{({(x - 5)}^{2} {(x + 3)}^{2})}^{\frac{1}{2}}})$

Step 4.2

Apply the product rule to ${(x - 5)}^{2} {(x + 3)}^{2}$ .

$\log (\frac{{(x^{4})}^{\frac{1}{2}}}{{({(x - 5)}^{2})}^{\frac{1}{2}} {({(x + 3)}^{2})}^{\frac{1}{2}}})$

Step 5

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

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Step 5.1

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

$\log (\frac{x^{4 (\frac{1}{2})}}{{({(x - 5)}^{2})}^{\frac{1}{2}} {({(x + 3)}^{2})}^{\frac{1}{2}}})$

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Factor $2$ out of $4$ .

$\log (\frac{x^{2 (2) \frac{1}{2}}}{{({(x - 5)}^{2})}^{\frac{1}{2}} {({(x + 3)}^{2})}^{\frac{1}{2}}})$

Step 5.2.2

Cancel the common factor.

$\log (\frac{x^{2 \cdot 2 \frac{1}{2}}}{{({(x - 5)}^{2})}^{\frac{1}{2}} {({(x + 3)}^{2})}^{\frac{1}{2}}})$

Step 5.2.3

Rewrite the expression.

$\log (\frac{x^{2}}{{({(x - 5)}^{2})}^{\frac{1}{2}} {({(x + 3)}^{2})}^{\frac{1}{2}}})$

Step 6

Simplify the denominator.

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Step 6.1

Multiply the exponents in ${({(x - 5)}^{2})}^{\frac{1}{2}}$ .

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Step 6.1.1

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

$\log (\frac{x^{2}}{{(x - 5)}^{2 (\frac{1}{2})} {({(x + 3)}^{2})}^{\frac{1}{2}}})$

Step 6.1.2

Cancel the common factor of $2$ .

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Step 6.1.2.1

Cancel the common factor.

$\log (\frac{x^{2}}{{(x - 5)}^{2 (\frac{1}{2})} {({(x + 3)}^{2})}^{\frac{1}{2}}})$

Step 6.1.2.2

Rewrite the expression.

$\log (\frac{x^{2}}{{(x - 5)}^{1} {({(x + 3)}^{2})}^{\frac{1}{2}}})$

Step 6.2

Simplify.

$\log (\frac{x^{2}}{(x - 5) {({(x + 3)}^{2})}^{\frac{1}{2}}})$

Step 6.3

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

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Step 6.3.1

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

$\log (\frac{x^{2}}{(x - 5) {(x + 3)}^{2 (\frac{1}{2})}})$

Step 6.3.2

Cancel the common factor of $2$ .

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Step 6.3.2.1

Cancel the common factor.

$\log (\frac{x^{2}}{(x - 5) {(x + 3)}^{2 (\frac{1}{2})}})$

Step 6.3.2.2

Rewrite the expression.

$\log (\frac{x^{2}}{(x - 5) {(x + 3)}^{1}})$

Step 6.4

Simplify.

$\log (\frac{x^{2}}{(x - 5) (x + 3)})$