Precalculus Examples

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

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{243}$ as $243^{\frac{1}{2}}$ .

$3 \log (3^{3 \cdot 243^{\frac{1}{2}}})$

Step 2

Expand $\log (3^{3 \cdot 243^{\frac{1}{2}}})$ by moving $3 \cdot 243^{\frac{1}{2}}$ outside the logarithm.

$3 (3 \cdot 243^{\frac{1}{2}} \log (3))$

Step 3

Multiply $3 \cdot 243^{\frac{1}{2}}$ .

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Step 3.1

Rewrite $243$ as $3^{5}$ .

$3 (3 \cdot {(3^{5})}^{\frac{1}{2}} \log (3))$

Step 3.2

Multiply the exponents in ${(3^{5})}^{\frac{1}{2}}$ .

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Step 3.2.1

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

$3 (3 \cdot 3^{5 (\frac{1}{2})} \log (3))$

Step 3.2.2

Combine $5$ and $\frac{1}{2}$ .

$3 (3 \cdot 3^{\frac{5}{2}} \log (3))$

Step 3.3

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

$3 (3^{1 + \frac{5}{2}} \log (3))$

Step 3.4

Write $1$ as a fraction with a common denominator.

$3 (3^{\frac{2}{2} + \frac{5}{2}} \log (3))$

Step 3.5

Combine the numerators over the common denominator.

$3 (3^{\frac{2 + 5}{2}} \log (3))$

Step 3.6

Add $2$ and $5$ .

$3 (3^{\frac{7}{2}} \log (3))$

Step 4

Multiply $3$ by $3^{\frac{7}{2}}$ by adding the exponents.

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Step 4.1

Move $3^{\frac{7}{2}}$ .

$3^{\frac{7}{2}} \cdot 3 \log (3)$

Step 4.2

Multiply $3^{\frac{7}{2}}$ by $3$ .

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Step 4.2.1

Raise $3$ to the power of $1$ .

$3^{\frac{7}{2}} \cdot 3^{1} \log (3)$

Step 4.2.2

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

$3^{\frac{7}{2} + 1} \log (3)$

Step 4.3

Write $1$ as a fraction with a common denominator.

$3^{\frac{7}{2} + \frac{2}{2}} \log (3)$

Step 4.4

Combine the numerators over the common denominator.

$3^{\frac{7 + 2}{2}} \log (3)$

Step 4.5

Add $7$ and $2$ .

$3^{\frac{9}{2}} \log (3)$