Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $\frac{5 (\log_{2} (x))}{6}$ as $\frac{5}{6} \log_{2} (x)$ .

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

Step 1.2

Simplify $\frac{5}{6} \log_{2} (x)$ by moving $\frac{5}{6}$ inside the logarithm.

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

Step 1.3

Rewrite $\frac{2 (\log_{2} (y))}{3}$ as $\frac{2}{3} \log_{2} (y)$ .

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

Step 1.4

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

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

Step 1.5

Rewrite $\frac{\log_{2} (x)}{2}$ as $\frac{1}{2} \log_{2} (x)$ .

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

Step 1.6

Simplify $\frac{1}{2} \log_{2} (x)$ by moving $\frac{1}{2}$ inside the logarithm.

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

Step 2

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

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

Step 3

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

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

Step 4

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

$\log_{2} (\frac{\frac{x^{\frac{5}{6}} y^{\frac{2}{3}}}{x^{\frac{1}{2}}}}{y})$

Step 5

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{5}{6}} y^{\frac{2}{3}}$ .

$\log_{2} (\frac{\frac{x^{\frac{1}{2}} (x^{\frac{2}{6}} y^{\frac{2}{3}})}{x^{\frac{1}{2}}}}{y})$

Step 6

Cancel the common factors.

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Step 6.1

Multiply by $1$ .

$\log_{2} (\frac{\frac{x^{\frac{1}{2}} (x^{\frac{2}{6}} y^{\frac{2}{3}})}{x^{\frac{1}{2}} \cdot 1}}{y})$

Step 6.2

Cancel the common factor.

$\log_{2} (\frac{\frac{x^{\frac{1}{2}} (x^{\frac{2}{6}} y^{\frac{2}{3}})}{x^{\frac{1}{2}} \cdot 1}}{y})$

Step 6.3

Rewrite the expression.

$\log_{2} (\frac{\frac{x^{\frac{2}{6}} y^{\frac{2}{3}}}{1}}{y})$

Step 6.4

Divide $x^{\frac{2}{6}} y^{\frac{2}{3}}$ by $1$ .

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

Step 7

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

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Step 7.1

Factor $2$ out of $2$ .

$\log_{2} (\frac{x^{\frac{2 (1)}{6}} y^{\frac{2}{3}}}{y})$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $2$ out of $6$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 8

Factor $y$ out of $x^{\frac{1}{3}} y^{\frac{2}{3}}$ .

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

Step 9

Cancel the common factors.

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Step 9.1

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

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

Step 9.2

Factor $y$ out of $y^{1}$ .

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

Step 9.3

Cancel the common factor.

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

Step 9.4

Rewrite the expression.

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

Step 9.5

Divide $x^{\frac{1}{3}} y^{- \frac{1}{3}}$ by $1$ .

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

Step 10

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 11

Combine $x^{\frac{1}{3}}$ and $\frac{1}{y^{\frac{1}{3}}}$ .

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