Find the Exact Value log base 3 of square root of 27+ log base 3 of 81

$\log_{3} (\sqrt{27}) + \log_{3} (81)$

Step 1

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

$\log_{3} (\sqrt{27} \cdot 81)$

Step 2

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$\log_{3} (\sqrt{9 (3)} \cdot 81)$

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

$\log_{3} (\sqrt{3^{2} \cdot 3} \cdot 81)$

Step 3

Pull terms out from under the radical.

$\log_{3} (3 \sqrt{3} \cdot 81)$

Step 4

Multiply $81$ by $3$ .

$\log_{3} (243 \sqrt{3})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\log_{3} (243 \sqrt{3})$

Decimal Form:

$5.5$