Precalculus Examples

$\ln (6) - (3 \ln (x) + 2 \ln (3))$

Step 1

Simplify each term.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$\ln (6) - (\ln (x^{3}) + 2 \ln (3))$

Step 1.1.2

Simplify $2 \ln (3)$ by moving $2$ inside the logarithm.

$\ln (6) - (\ln (x^{3}) + \ln (3^{2}))$

Step 1.1.3

Raise $3$ to the power of $2$ .

$\ln (6) - (\ln (x^{3}) + \ln (9))$

Step 1.2

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

$\ln (6) - \ln (x^{3} \cdot 9)$

Step 1.3

Move $9$ to the left of $x^{3}$ .

$\ln (6) - \ln (9 x^{3})$

Step 2

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

$\ln (\frac{6}{9 x^{3}})$

Step 3

Cancel the common factor of $6$ and $9$ .

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Step 3.1

Factor $3$ out of $6$ .

$\ln (\frac{3 (2)}{9 x^{3}})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $3$ out of $9 x^{3}$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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