Precalculus Examples

$\log_{2} (5 x^{2} y^{3}) - \log_{2} (20 x^{4} y) + \log_{2} (2 x y^{6})$

Step 1

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

$\log_{2} (\frac{5 x^{2} y^{3}}{20 x^{4} y}) + \log_{2} (2 x y^{6})$

Step 2

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

$\log_{2} (\frac{5 x^{2} y^{3}}{20 x^{4} y} (2 x y^{6}))$

Step 3

Rewrite using the commutative property of multiplication.

$\log_{2} (2 \frac{5 x^{2} y^{3}}{20 x^{4} y} (x y^{6}))$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $20 x^{4} y$ .

$\log_{2} (2 \frac{5 x^{2} y^{3}}{2 (10 x^{4} y)} (x y^{6}))$

Step 4.2

Cancel the common factor.

$\log_{2} (2 \frac{5 x^{2} y^{3}}{2 (10 x^{4} y)} (x y^{6}))$

Step 4.3

Rewrite the expression.

$\log_{2} (\frac{5 x^{2} y^{3}}{10 x^{4} y} (x y^{6}))$

Step 5

Cancel the common factor of $x y$ .

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Step 5.1

Factor $x y$ out of $10 x^{4} y$ .

$\log_{2} (\frac{5 x^{2} y^{3}}{x y (10 x^{3})} (x y^{6}))$

Step 5.2

Factor $x y$ out of $x y^{6}$ .

$\log_{2} (\frac{5 x^{2} y^{3}}{x y (10 x^{3})} (x y (y^{5})))$

Step 5.3

Cancel the common factor.

$\log_{2} (\frac{5 x^{2} y^{3}}{x y (10 x^{3})} (x y \cdot y^{5}))$

Step 5.4

Rewrite the expression.

$\log_{2} (\frac{5 x^{2} y^{3}}{10 x^{3}} y^{5})$

Step 6

Combine $\frac{5 x^{2} y^{3}}{10 x^{3}}$ and $y^{5}$ .

$\log_{2} (\frac{5 x^{2} y^{3} y^{5}}{10 x^{3}})$

Step 7

Multiply $y^{3}$ by $y^{5}$ by adding the exponents.

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Step 7.1

Move $y^{5}$ .

$\log_{2} (\frac{5 x^{2} (y^{5} y^{3})}{10 x^{3}})$

Step 7.2

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

$\log_{2} (\frac{5 x^{2} y^{5 + 3}}{10 x^{3}})$

Step 7.3

Add $5$ and $3$ .

$\log_{2} (\frac{5 x^{2} y^{8}}{10 x^{3}})$

Step 8

Cancel the common factor of $5$ and $10$ .

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Step 8.1

Factor $5$ out of $5 x^{2} y^{8}$ .

$\log_{2} (\frac{5 (x^{2} y^{8})}{10 x^{3}})$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $5$ out of $10 x^{3}$ .

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

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

Step 9

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 9.1

Factor $x^{2}$ out of $x^{2} y^{8}$ .

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

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $x^{2}$ out of $2 x^{3}$ .

$\log_{2} (\frac{x^{2} (y^{8})}{x^{2} (2 x)})$

Step 9.2.2

Cancel the common factor.

$\log_{2} (\frac{x^{2} y^{8}}{x^{2} (2 x)})$

Step 9.2.3

Rewrite the expression.

$\log_{2} (\frac{y^{8}}{2 x})$