Precalculus Examples

$5 \log_{3} (x + 2) + 2 \log_{3} (4 x) - \log_{3} (8 x^{5})$

Step 1

Simplify each term.

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Step 1.1

Simplify $5 \log_{3} (x + 2)$ by moving $5$ inside the logarithm.

$\log_{3} ({(x + 2)}^{5}) + 2 \log_{3} (4 x) - \log_{3} (8 x^{5})$

Step 1.2

Simplify $2 \log_{3} (4 x)$ by moving $2$ inside the logarithm.

$\log_{3} ({(x + 2)}^{5}) + \log_{3} ({(4 x)}^{2}) - \log_{3} (8 x^{5})$

Step 1.3

Apply the product rule to $4 x$ .

$\log_{3} ({(x + 2)}^{5}) + \log_{3} (4^{2} x^{2}) - \log_{3} (8 x^{5})$

Step 1.4

Raise $4$ to the power of $2$ .

$\log_{3} ({(x + 2)}^{5}) + \log_{3} (16 x^{2}) - \log_{3} (8 x^{5})$

Step 2

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

$\log_{3} ({(x + 2)}^{5} (16 x^{2})) - \log_{3} (8 x^{5})$

Step 3

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

$\log_{3} (\frac{{(x + 2)}^{5} (16 x^{2})}{8 x^{5}})$

Step 4

Cancel the common factor of $16$ and $8$ .

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Step 4.1

Factor $8$ out of ${(x + 2)}^{5} (16 x^{2})$ .

$\log_{3} (\frac{8 ({(x + 2)}^{5} (2 x^{2}))}{8 x^{5}})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $8$ out of $8 x^{5}$ .

$\log_{3} (\frac{8 ({(x + 2)}^{5} (2 x^{2}))}{8 (x^{5})})$

Step 4.2.2

Cancel the common factor.

$\log_{3} (\frac{8 ({(x + 2)}^{5} (2 x^{2}))}{8 x^{5}})$

Step 4.2.3

Rewrite the expression.

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

Step 5

Cancel the common factor of $x^{2}$ and $x^{5}$ .

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Step 5.1

Factor $x^{2}$ out of ${(x + 2)}^{5} (2 x^{2})$ .

$\log_{3} (\frac{x^{2} ({(x + 2)}^{5} \cdot (2))}{x^{5}})$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $x^{2}$ out of $x^{5}$ .

$\log_{3} (\frac{x^{2} ({(x + 2)}^{5} \cdot (2))}{x^{2} x^{3}})$

Step 5.2.2

Cancel the common factor.

$\log_{3} (\frac{x^{2} ({(x + 2)}^{5} \cdot (2))}{x^{2} x^{3}})$

Step 5.2.3

Rewrite the expression.

$\log_{3} (\frac{{(x + 2)}^{5} \cdot (2)}{x^{3}})$

Step 6

Move $2$ to the left of ${(x + 2)}^{5}$ .

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