Precalculus Examples

$3 \log ({(3)}^{3 \sqrt{243}})$

Step 1

Rewrite $243$ as $9^{2} \cdot 3$ .

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Step 1.1

Factor $81$ out of $243$ .

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

Step 1.2

Rewrite $81$ as $9^{2}$ .

$3 \log (3^{3 \sqrt{9^{2} \cdot 3}})$

Step 2

Pull terms out from under the radical.

$3 \log (3^{3 (9 \sqrt{3})})$

Step 3

Multiply $9$ by $3$ .

$3 \log (3^{27 \sqrt{3}})$

Step 4

Simplify $3 \log (3^{27 \sqrt{3}})$ by moving $3$ inside the logarithm.

$\log ({(3^{27 \sqrt{3}})}^{3})$

Step 5

Multiply the exponents in ${(3^{27 \sqrt{3}})}^{3}$ .

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Step 5.1

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

$\log (3^{27 \sqrt{3} \cdot 3})$

Step 5.2

Multiply $3$ by $27$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$66.93825861 \dots$