Precalculus Examples

$\ln (2 x) + 3 \ln (y) - \ln (x^{2 - 1})$

Step 1

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

$3 \ln (y) + \ln (\frac{2 x}{x^{2 - 1}})$

Step 2

Simplify each term.

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Step 2.1

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

$\ln (y^{3}) + \ln (\frac{2 x}{x^{2 - 1}})$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Subtract $1$ from $2$ .

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

Step 2.2.2

Simplify.

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

Step 2.3

Cancel the common factor of $x$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Divide $2$ by $1$ .

$\ln (y^{3}) + \ln (2)$

Step 3

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

$\ln (y^{3} \cdot 2)$

Step 4

Move $2$ to the left of $y^{3}$ .

$\ln (2 y^{3})$