Basic Math Examples

$\log (x^{2}) - \log (x y) + 4 \log (y)$

Step 1

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

$\log (\frac{x^{2}}{x y}) + 4 \log (y)$

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

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

$\log (\frac{x \cdot x}{x y}) + 4 \log (y)$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $x$ out of $x y$ .

$\log (\frac{x \cdot x}{x (y)}) + 4 \log (y)$

Step 2.1.2.2

Cancel the common factor.

$\log (\frac{x \cdot x}{x y}) + 4 \log (y)$

Step 2.1.2.3

Rewrite the expression.

$\log (\frac{x}{y}) + 4 \log (y)$

Step 2.2

Simplify $4 \log (y)$ by moving $4$ inside the logarithm.

$\log (\frac{x}{y}) + \log (y^{4})$

Step 3

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

$\log (\frac{x}{y} y^{4})$

Step 4

Cancel the common factor of $y$ .

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Step 4.1

Factor $y$ out of $y^{4}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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