Precalculus Examples

$3 \ln (x) - 5 \ln (x y)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \ln (x)$ by moving $3$ inside the logarithm.

$\ln (x^{3}) - 5 \ln (x y)$

Step 1.2

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

$\ln (x^{3}) - \ln ({(x y)}^{5})$

Step 1.3

Apply the product rule to $x y$ .

$\ln (x^{3}) - \ln (x^{5} y^{5})$

Step 2

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

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

Step 3

Cancel the common factor of $x^{3}$ and $x^{5}$ .

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Step 3.1

Multiply by $1$ .

$\ln (\frac{x^{3} \cdot 1}{x^{5} y^{5}})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x^{3}$ out of $x^{5} y^{5}$ .

$\ln (\frac{x^{3} \cdot 1}{x^{3} (x^{2} y^{5})})$

Step 3.2.2

Cancel the common factor.

$\ln (\frac{x^{3} \cdot 1}{x^{3} (x^{2} y^{5})})$

Step 3.2.3

Rewrite the expression.

$\ln (\frac{1}{x^{2} y^{5}})$