Precalculus Examples

$\ln (2) + \ln (6) - \frac{1}{2} \cdot \ln (9)$

Step 1

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

$\ln (2 \cdot 6) - \frac{1}{2} \cdot \ln (9)$

Step 2

Simplify each term.

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Step 2.1

Multiply $2$ by $6$ .

$\ln (12) - \frac{1}{2} \cdot \ln (9)$

Step 2.2

Multiply $- \frac{1}{2} \ln (9)$ .

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Step 2.2.1

Reorder $\ln (9)$ and $\frac{1}{2}$ .

$\ln (12) - (\frac{1}{2} \ln (9))$

Step 2.2.2

Simplify $\frac{1}{2} \ln (9)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln (12) - \ln (9^{\frac{1}{2}})$

Step 2.3

Rewrite $9$ as $3^{2}$ .

$\ln (12) - \ln ({(3^{2})}^{\frac{1}{2}})$

Step 2.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (12) - \ln (3^{2 (\frac{1}{2})})$

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Cancel the common factor.

$\ln (12) - \ln (3^{2 (\frac{1}{2})})$

Step 2.5.2

Rewrite the expression.

$\ln (12) - \ln (3^{1})$

Step 2.6

Evaluate the exponent.

$\ln (12) - \ln (3)$

Step 3

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

$\ln (\frac{12}{3})$

Step 4

Divide $12$ by $3$ .

$\ln (4)$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\ln (4)$

Decimal Form:

$1.38629436 \dots$