Precalculus Examples

$2 \log (\frac{y^{4}}{x}) - 3 \log (y) + \frac{1}{2} \cdot \log (x^{4} y^{2})$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log (\frac{y^{4}}{x})$ by moving $2$ inside the logarithm.

$\log ({(\frac{y^{4}}{x})}^{2}) - 3 \log (y) + \frac{1}{2} \cdot \log (x^{4} y^{2})$

Step 1.2

Apply the product rule to $\frac{y^{4}}{x}$ .

$\log (\frac{{(y^{4})}^{2}}{x^{2}}) - 3 \log (y) + \frac{1}{2} \cdot \log (x^{4} y^{2})$

Step 1.3

Multiply the exponents in ${(y^{4})}^{2}$ .

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Step 1.3.1

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

$\log (\frac{y^{4 \cdot 2}}{x^{2}}) - 3 \log (y) + \frac{1}{2} \cdot \log (x^{4} y^{2})$

Step 1.3.2

Multiply $4$ by $2$ .

$\log (\frac{y^{8}}{x^{2}}) - 3 \log (y) + \frac{1}{2} \cdot \log (x^{4} y^{2})$

Step 1.4

Simplify $- 3 \log (y)$ by moving $3$ inside the logarithm.

$\log (\frac{y^{8}}{x^{2}}) - \log (y^{3}) + \frac{1}{2} \cdot \log (x^{4} y^{2})$

Step 1.5

Simplify $\frac{1}{2} \log (x^{4} y^{2})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log (\frac{y^{8}}{x^{2}}) - \log (y^{3}) + \log ({(x^{4} y^{2})}^{\frac{1}{2}})$

Step 1.6

Apply the product rule to $x^{4} y^{2}$ .

$\log (\frac{y^{8}}{x^{2}}) - \log (y^{3}) + \log ({(x^{4})}^{\frac{1}{2}} {(y^{2})}^{\frac{1}{2}})$

Step 1.7

Multiply the exponents in ${(x^{4})}^{\frac{1}{2}}$ .

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Step 1.7.1

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

$\log (\frac{y^{8}}{x^{2}}) - \log (y^{3}) + \log (x^{4 (\frac{1}{2})} {(y^{2})}^{\frac{1}{2}})$

Step 1.7.2

Cancel the common factor of $2$ .

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Step 1.7.2.1

Factor $2$ out of $4$ .

$\log (\frac{y^{8}}{x^{2}}) - \log (y^{3}) + \log (x^{2 (2) \frac{1}{2}} {(y^{2})}^{\frac{1}{2}})$

Step 1.7.2.2

Cancel the common factor.

$\log (\frac{y^{8}}{x^{2}}) - \log (y^{3}) + \log (x^{2 \cdot 2 \frac{1}{2}} {(y^{2})}^{\frac{1}{2}})$

Step 1.7.2.3

Rewrite the expression.

$\log (\frac{y^{8}}{x^{2}}) - \log (y^{3}) + \log (x^{2} {(y^{2})}^{\frac{1}{2}})$

Step 1.8

Multiply the exponents in ${(y^{2})}^{\frac{1}{2}}$ .

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Step 1.8.1

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

$\log (\frac{y^{8}}{x^{2}}) - \log (y^{3}) + \log (x^{2} y^{2 (\frac{1}{2})})$

Step 1.8.2

Cancel the common factor of $2$ .

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Step 1.8.2.1

Cancel the common factor.

$\log (\frac{y^{8}}{x^{2}}) - \log (y^{3}) + \log (x^{2} y^{2 (\frac{1}{2})})$

Step 1.8.2.2

Rewrite the expression.

$\log (\frac{y^{8}}{x^{2}}) - \log (y^{3}) + \log (x^{2} y^{1})$

Step 1.9

Simplify.

$\log (\frac{y^{8}}{x^{2}}) - \log (y^{3}) + \log (x^{2} y)$

Step 2

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

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

Step 3

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

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

Step 4

Cancel the common factor of $y$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Cancel the common factor.

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

Step 4.4

Rewrite the expression.

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

Step 5

Combine $\frac{\frac{y^{8}}{x^{2}}}{y^{2}}$ and $x^{2}$ .

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

Step 6

Combine $\frac{y^{8}}{x^{2}}$ and $x^{2}$ .

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

Step 7

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

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Step 7.1

Cancel the common factor.

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

Step 7.2

Divide $y^{8}$ by $1$ .

$\log (\frac{y^{8}}{y^{2}})$

Step 8

Cancel the common factor of $y^{8}$ and $y^{2}$ .

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Step 8.1

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

$\log (\frac{y^{2} y^{6}}{y^{2}})$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Multiply by $1$ .

$\log (\frac{y^{2} y^{6}}{y^{2} \cdot 1})$

Step 8.2.2

Cancel the common factor.

$\log (\frac{y^{2} y^{6}}{y^{2} \cdot 1})$

Step 8.2.3

Rewrite the expression.

$\log (\frac{y^{6}}{1})$

Step 8.2.4

Divide $y^{6}$ by $1$ .

$\log (y^{6})$