Precalculus Examples

$5 \ln (x y) - 2 \ln (y z)$

Step 1

Simplify each term.

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Step 1.1

Simplify $5 \ln (x y)$ by moving $5$ inside the logarithm.

$\ln ({(x y)}^{5}) - 2 \ln (y z)$

Step 1.2

Apply the product rule to $x y$ .

$\ln (x^{5} y^{5}) - 2 \ln (y z)$

Step 1.3

Simplify $- 2 \ln (y z)$ by moving $2$ inside the logarithm.

$\ln (x^{5} y^{5}) - \ln ({(y z)}^{2})$

Step 1.4

Apply the product rule to $y z$ .

$\ln (x^{5} y^{5}) - \ln (y^{2} z^{2})$

Step 2

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

$\ln (\frac{x^{5} y^{5}}{y^{2} z^{2}})$

Step 3

Cancel the common factor of $y^{5}$ and $y^{2}$ .

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Step 3.1

Factor $y^{2}$ out of $x^{5} y^{5}$ .

$\ln (\frac{y^{2} (x^{5} y^{3})}{y^{2} z^{2}})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $y^{2}$ out of $y^{2} z^{2}$ .

$\ln (\frac{y^{2} (x^{5} y^{3})}{y^{2} (z^{2})})$

Step 3.2.2

Cancel the common factor.

$\ln (\frac{y^{2} (x^{5} y^{3})}{y^{2} z^{2}})$

Step 3.2.3

Rewrite the expression.

$\ln (\frac{x^{5} y^{3}}{z^{2}})$