Precalculus Examples

$4 \log (x y) - 3 \log (y z)$

Step 1

Simplify each term.

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Step 1.1

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

$\log ({(x y)}^{4}) - 3 \log (y z)$

Step 1.2

Apply the product rule to $x y$ .

$\log (x^{4} y^{4}) - 3 \log (y z)$

Step 1.3

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

$\log (x^{4} y^{4}) - \log ({(y z)}^{3})$

Step 1.4

Apply the product rule to $y z$ .

$\log (x^{4} y^{4}) - \log (y^{3} z^{3})$

Step 2

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

$\log (\frac{x^{4} y^{4}}{y^{3} z^{3}})$

Step 3

Cancel the common factor of $y^{4}$ and $y^{3}$ .

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Step 3.1

Factor $y^{3}$ out of $x^{4} y^{4}$ .

$\log (\frac{y^{3} (x^{4} y)}{y^{3} z^{3}})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

$\log (\frac{y^{3} (x^{4} y)}{y^{3} (z^{3})})$

Step 3.2.2

Cancel the common factor.

$\log (\frac{y^{3} (x^{4} y)}{y^{3} z^{3}})$

Step 3.2.3

Rewrite the expression.

$\log (\frac{x^{4} y}{z^{3}})$