Precalculus Examples

$\frac{1}{2} \cdot \ln (3) + \ln (9)$

Step 1

Simplify $\frac{1}{2} \ln (3)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln (3^{\frac{1}{2}}) + \ln (9)$

Step 2

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

$\ln (3^{\frac{1}{2}} \cdot 9)$

Step 3

Multiply $3^{\frac{1}{2}} \cdot 9$ .

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Step 3.1

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

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

Step 3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.4

Combine $2$ and $\frac{2}{2}$ .

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $2$ by $2$ .

$\ln (3^{\frac{1 + 4}{2}})$

Step 3.6.2

Add $1$ and $4$ .

$\ln (3^{\frac{5}{2}})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\ln (3^{\frac{5}{2}})$

Decimal Form:

$2.74653072 \dots$