Precalculus Examples

$\log_{z} (x y^{2}) + 2 \log_{z} (\frac{x}{y}) - 3 \log_{z} (y x^{\frac{2}{3}})$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{z} (\frac{x}{y})$ by moving $2$ inside the logarithm.

$\log_{z} (x y^{2}) + \log_{z} ({(\frac{x}{y})}^{2}) - 3 \log_{z} (y x^{\frac{2}{3}})$

Step 1.2

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

$\log_{z} (x y^{2}) + \log_{z} (\frac{x^{2}}{y^{2}}) - 3 \log_{z} (y x^{\frac{2}{3}})$

Step 1.3

Simplify $- 3 \log_{z} (y x^{\frac{2}{3}})$ by moving $3$ inside the logarithm.

$\log_{z} (x y^{2}) + \log_{z} (\frac{x^{2}}{y^{2}}) - \log_{z} ({(y x^{\frac{2}{3}})}^{3})$

Step 1.4

Apply the product rule to $y x^{\frac{2}{3}}$ .

$\log_{z} (x y^{2}) + \log_{z} (\frac{x^{2}}{y^{2}}) - \log_{z} (y^{3} {(x^{\frac{2}{3}})}^{3})$

Step 1.5

Multiply the exponents in ${(x^{\frac{2}{3}})}^{3}$ .

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Step 1.5.1

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

$\log_{z} (x y^{2}) + \log_{z} (\frac{x^{2}}{y^{2}}) - \log_{z} (y^{3} x^{\frac{2}{3} \cdot 3})$

Step 1.5.2

Cancel the common factor of $3$ .

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Step 1.5.2.1

Cancel the common factor.

$\log_{z} (x y^{2}) + \log_{z} (\frac{x^{2}}{y^{2}}) - \log_{z} (y^{3} x^{\frac{2}{3} \cdot 3})$

Step 1.5.2.2

Rewrite the expression.

$\log_{z} (x y^{2}) + \log_{z} (\frac{x^{2}}{y^{2}}) - \log_{z} (y^{3} x^{2})$

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Cancel the common factors.

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Step 5.1

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

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

Combine.

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

Step 8

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 8.1

Move $y$ .

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

Step 8.2

Multiply $y$ by $y^{2}$ .

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Step 8.2.1

Raise $y$ to the power of $1$ .

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

Step 8.2.2

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

$\log_{z} (\frac{x^{2} \cdot 1}{y^{1 + 2} x})$

Step 8.3

Add $1$ and $2$ .

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

Step 9

Multiply $x^{2}$ by $1$ .

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

Step 10

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

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Step 10.1

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

$\log_{z} (\frac{x \cdot x}{y^{3} x})$

Step 10.2

Cancel the common factors.

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Step 10.2.1

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

$\log_{z} (\frac{x \cdot x}{x y^{3}})$

Step 10.2.2

Cancel the common factor.

$\log_{z} (\frac{x \cdot x}{x y^{3}})$

Step 10.2.3

Rewrite the expression.

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