Precalculus Examples

$\frac{1}{3} \cdot \log ({(x + 3)}^{2}) + \log (x^{2}) - \log (x^{2} - 2 x - 15)$

Step 1

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

$\frac{1}{3} \cdot \log ({(x + 3)}^{2}) + \log (\frac{x^{2}}{x^{2} - 2 x - 15})$

Step 2

Simplify each term.

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Step 2.1

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

$\log ({({(x + 3)}^{2})}^{\frac{1}{3}}) + \log (\frac{x^{2}}{x^{2} - 2 x - 15})$

Step 2.2

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

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Step 2.2.1

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

$\log ({(x + 3)}^{2 (\frac{1}{3})}) + \log (\frac{x^{2}}{x^{2} - 2 x - 15})$

Step 2.2.2

Combine $2$ and $\frac{1}{3}$ .

$\log ({(x + 3)}^{\frac{2}{3}}) + \log (\frac{x^{2}}{x^{2} - 2 x - 15})$

Step 2.3

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

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Step 2.3.1

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 2.3.2

Write the factored form using these integers.

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

Step 3

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

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

Step 4

Combine ${(x + 3)}^{\frac{2}{3}}$ and $\frac{x^{2}}{(x - 5) (x + 3)}$ .

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

Step 5

Move ${(x + 3)}^{\frac{2}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 6

Multiply $x + 3$ by ${(x + 3)}^{- \frac{2}{3}}$ by adding the exponents.

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Step 6.1

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

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

Step 6.2

Multiply ${(x + 3)}^{- \frac{2}{3}}$ by $x + 3$ .

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Step 6.2.1

Raise $x + 3$ to the power of $1$ .

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

Step 6.2.2

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

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

Step 6.3

Write $1$ as a fraction with a common denominator.

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 6.5

Add $- 2$ and $3$ .

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

Step 7

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

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