Precalculus Examples

$\log (\frac{x^{2}}{y^{3}}) + 5 \log (y) - 6 \log (\sqrt{x y})$

Step 1

Simplify each term.

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Step 1.1

Simplify $5 \log (y)$ by moving $5$ inside the logarithm.

$\log (\frac{x^{2}}{y^{3}}) + \log (y^{5}) - 6 \log (\sqrt{x y})$

Step 1.2

Simplify $- 6 \log (\sqrt{x y})$ by moving $6$ inside the logarithm.

$\log (\frac{x^{2}}{y^{3}}) + \log (y^{5}) - \log ({\sqrt{x y}}^{6})$

Step 1.3

Rewrite ${\sqrt{x y}}^{6}$ as ${(x y)}^{3}$ .

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Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x y}$ as ${(x y)}^{\frac{1}{2}}$ .

$\log (\frac{x^{2}}{y^{3}}) + \log (y^{5}) - \log ({({(x y)}^{\frac{1}{2}})}^{6})$

Step 1.3.2

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

$\log (\frac{x^{2}}{y^{3}}) + \log (y^{5}) - \log ({(x y)}^{\frac{1}{2} \cdot 6})$

Step 1.3.3

Combine $\frac{1}{2}$ and $6$ .

$\log (\frac{x^{2}}{y^{3}}) + \log (y^{5}) - \log ({(x y)}^{\frac{6}{2}})$

Step 1.3.4

Cancel the common factor of $6$ and $2$ .

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Step 1.3.4.1

Factor $2$ out of $6$ .

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

Step 1.3.4.2

Cancel the common factors.

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Step 1.3.4.2.1

Factor $2$ out of $2$ .

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

Step 1.3.4.2.2

Cancel the common factor.

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

Step 1.3.4.2.3

Rewrite the expression.

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

Step 1.3.4.2.4

Divide $3$ by $1$ .

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

Step 1.4

Apply the product rule to $x y$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Cancel the common factor of $y^{5}$ and $y^{3}$ .

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Step 4.1

Factor $y^{3}$ out of $\frac{x^{2}}{y^{3}} y^{5}$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Reduce the expression $\frac{x^{2} y^{2}}{y^{3}}$ by cancelling the common factors.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Cancel the common factor.

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

Step 6.4

Rewrite the expression.

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

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{x^{2}}{y} \cdot \frac{1}{x^{3}})$

Step 8

Combine.

$\log (\frac{x^{2} \cdot 1}{y x^{3}})$

Step 9

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

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Step 9.1

Factor $x^{2}$ out of $x^{2} \cdot 1$ .

$\log (\frac{x^{2} (1)}{y x^{3}})$

Step 9.2

Cancel the common factors.

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Step 9.2.1

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

$\log (\frac{x^{2} (1)}{x^{2} (y x)})$

Step 9.2.2

Cancel the common factor.

$\log (\frac{x^{2} \cdot 1}{x^{2} (y x)})$

Step 9.2.3

Rewrite the expression.

$\log (\frac{1}{y x})$