Precalculus Examples

$\log_{2} (x y) - 3 \log_{2} (y) - 0.5 \log_{2} (x)$

Step 1

Simplify each term.

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Step 1.1

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

$\log_{2} (x y) - \log_{2} (y^{3}) - 0.5 \log_{2} (x)$

Step 1.2

Simplify $- 0.5 \log_{2} (x)$ by moving $0.5$ inside the logarithm.

$\log_{2} (x y) - \log_{2} (y^{3}) - \log_{2} (x^{0.5})$

Step 2

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

$\log_{2} (\frac{x y}{y^{3}}) - \log_{2} (x^{0.5})$

Step 3

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

$\log_{2} (\frac{\frac{x y}{y^{3}}}{x^{0.5}})$

Step 4

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

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Step 4.1

Factor $y$ out of $x y$ .

$\log_{2} (\frac{\frac{y x}{y^{3}}}{x^{0.5}})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

$\log_{2} (\frac{\frac{y x}{y \cdot y^{2}}}{x^{0.5}})$

Step 4.2.2

Cancel the common factor.

$\log_{2} (\frac{\frac{y x}{y \cdot y^{2}}}{x^{0.5}})$

Step 4.2.3

Rewrite the expression.

$\log_{2} (\frac{\frac{x}{y^{2}}}{x^{0.5}})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log_{2} (\frac{x}{y^{2}} \cdot \frac{1}{x^{0.5}})$

Step 6

Combine.

$\log_{2} (\frac{x \cdot 1}{y^{2} x^{0.5}})$

Step 7

Move $x^{0.5}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\log_{2} (\frac{x \cdot x^{- 0.5}}{y^{2}})$

Step 8

Multiply $x$ by $x^{- 0.5}$ by adding the exponents.

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Step 8.1

Multiply $x$ by $x^{- 0.5}$ .

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Step 8.1.1

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

$\log_{2} (\frac{x^{1} x^{- 0.5}}{y^{2}})$

Step 8.1.2

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

$\log_{2} (\frac{x^{1 - 0.5}}{y^{2}})$

Step 8.2

Subtract $0.5$ from $1$ .

$\log_{2} (\frac{x^{0.5}}{y^{2}})$