Precalculus Examples

$\log ({(\sqrt{3})}^{81})$

Step 1

Rewrite ${\sqrt{3}}^{81}$ as $\sqrt{3^{81}}$ .

$\log (\sqrt{3^{81}})$

Step 2

Rewrite $3^{81}$ as ${(3^{40})}^{2} \cdot 3$ .

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Step 2.1

Factor out $3^{80}$ .

$\log (\sqrt{3^{80} \cdot 3})$

Step 2.2

Rewrite $3^{80}$ as ${(3^{40})}^{2}$ .

$\log (\sqrt{{(3^{40})}^{2} \cdot 3})$

$\log (\sqrt{{(3^{40})}^{2} \cdot 3})$

Step 3

Pull terms out from under the radical.

$\log (3^{40} \sqrt{3})$

Step 4

Raise $3$ to the power of $40$ .

$\log (12157665459056900000 \sqrt{3})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\log (12157665459056900000 \sqrt{3})$

Decimal Form:

$19.32341081 \dots$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$