Precalculus Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

Rewrite using the commutative property of multiplication.

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

Step 2.2

Multiply $3$ by $3$ .

$\ln (9 x y) - 3 \ln (x)$

Step 2.3

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

$\ln (9 x y) - \ln (x^{3})$

Step 3

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

$\ln (\frac{9 x y}{x^{3}})$

Step 4

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

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Step 4.1

Factor $x$ out of $9 x y$ .

$\ln (\frac{x (9 y)}{x^{3}})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

$\ln (\frac{x (9 y)}{x \cdot x^{2}})$

Step 4.2.2

Cancel the common factor.

$\ln (\frac{x (9 y)}{x \cdot x^{2}})$

Step 4.2.3

Rewrite the expression.

$\ln (\frac{9 y}{x^{2}})$