Precalculus Examples

$\log_{3} (x^{2 - 9}) - 2 \log_{3} (x + 3) + 3 \log_{3} (x)$

Step 1

Simplify each term.

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Step 1.1

Subtract $9$ from $2$ .

$\log_{3} (x^{- 7}) - 2 \log_{3} (x + 3) + 3 \log_{3} (x)$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\log_{3} (\frac{1}{x^{7}}) - 2 \log_{3} (x + 3) + 3 \log_{3} (x)$

Step 1.3

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

$\log_{3} (\frac{1}{x^{7}}) - \log_{3} ({(x + 3)}^{2}) + 3 \log_{3} (x)$

Step 1.4

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

$\log_{3} (\frac{1}{x^{7}}) - \log_{3} ({(x + 3)}^{2}) + \log_{3} (x^{3})$

Step 2

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

$\log_{3} (\frac{\frac{1}{x^{7}}}{{(x + 3)}^{2}}) + \log_{3} (x^{3})$

Step 3

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

$\log_{3} (\frac{\frac{1}{x^{7}}}{{(x + 3)}^{2}} x^{3})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{3} (\frac{1}{x^{7}} \cdot \frac{1}{{(x + 3)}^{2}} x^{3})$

Step 5

Combine.

$\log_{3} (\frac{1 \cdot 1}{x^{7} {(x + 3)}^{2}} x^{3})$

Step 6

Cancel the common factor of $x^{3}$ .

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Step 6.1

Factor $x^{3}$ out of $x^{7} {(x + 3)}^{2}$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Multiply $1$ by $1$ .

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