Algebra Examples

$3 \ln (9) - 2 \ln (6) - \ln (3)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \ln (9)$ by moving $3$ inside the logarithm.

$\ln (9^{3}) - 2 \ln (6) - \ln (3)$

Step 1.2

Raise $9$ to the power of $3$ .

$\ln (729) - 2 \ln (6) - \ln (3)$

Step 1.3

Simplify $- 2 \ln (6)$ by moving $2$ inside the logarithm.

$\ln (729) - \ln (6^{2}) - \ln (3)$

Step 1.4

Raise $6$ to the power of $2$ .

$\ln (729) - \ln (36) - \ln (3)$

Step 2

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

$\ln (\frac{729}{36}) - \ln (3)$

Step 3

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

$\ln (\frac{\frac{729}{36}}{3})$

Step 4

Cancel the common factor of $729$ and $36$ .

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Step 4.1

Factor $9$ out of $729$ .

$\ln (\frac{\frac{9 (81)}{36}}{3})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $9$ out of $36$ .

$\ln (\frac{\frac{9 \cdot 81}{9 \cdot 4}}{3})$

Step 4.2.2

Cancel the common factor.

$\ln (\frac{\frac{9 \cdot 81}{9 \cdot 4}}{3})$

Step 4.2.3

Rewrite the expression.

$\ln (\frac{\frac{81}{4}}{3})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{81}{4} \cdot \frac{1}{3})$

Step 6

Cancel the common factor of $3$ .

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Step 6.1

Factor $3$ out of $81$ .

$\ln (\frac{3 (27)}{4} \cdot \frac{1}{3})$

Step 6.2

Cancel the common factor.

$\ln (\frac{3 \cdot 27}{4} \cdot \frac{1}{3})$

Step 6.3

Rewrite the expression.

$\ln (\frac{27}{4})$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{27}{4})$

Decimal Form:

$1.90954250 \dots$