Precalculus Examples

$21 \log_{3} (\sqrt[3]{x}) + \log_{3} (9 x^{2}) - \log_{2} (9)$

Step 1

Simplify each term.

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Step 1.1

Simplify $21 \log_{3} (\sqrt[3]{x})$ by moving $21$ inside the logarithm.

$\log_{3} ({\sqrt[3]{x}}^{21}) + \log_{3} (9 x^{2}) - \log_{2} (9)$

Step 1.2

Rewrite ${\sqrt[3]{x}}^{21}$ as $x^{7}$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\log_{3} ({(x^{\frac{1}{3}})}^{21}) + \log_{3} (9 x^{2}) - \log_{2} (9)$

Step 1.2.2

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

$\log_{3} (x^{\frac{1}{3} \cdot 21}) + \log_{3} (9 x^{2}) - \log_{2} (9)$

Step 1.2.3

Combine $\frac{1}{3}$ and $21$ .

$\log_{3} (x^{\frac{21}{3}}) + \log_{3} (9 x^{2}) - \log_{2} (9)$

Step 1.2.4

Cancel the common factor of $21$ and $3$ .

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Step 1.2.4.1

Factor $3$ out of $21$ .

$\log_{3} (x^{\frac{3 \cdot 7}{3}}) + \log_{3} (9 x^{2}) - \log_{2} (9)$

Step 1.2.4.2

Cancel the common factors.

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Step 1.2.4.2.1

Factor $3$ out of $3$ .

$\log_{3} (x^{\frac{3 \cdot 7}{3 (1)}}) + \log_{3} (9 x^{2}) - \log_{2} (9)$

Step 1.2.4.2.2

Cancel the common factor.

$\log_{3} (x^{\frac{3 \cdot 7}{3 \cdot 1}}) + \log_{3} (9 x^{2}) - \log_{2} (9)$

Step 1.2.4.2.3

Rewrite the expression.

$\log_{3} (x^{\frac{7}{1}}) + \log_{3} (9 x^{2}) - \log_{2} (9)$

Step 1.2.4.2.4

Divide $7$ by $1$ .

$\log_{3} (x^{7}) + \log_{3} (9 x^{2}) - \log_{2} (9)$

Step 2

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

$\log_{3} (x^{7} (9 x^{2})) - \log_{2} (9)$

Step 3

Simplify each term.

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Step 3.1

Rewrite using the commutative property of multiplication.

$\log_{3} (9 x^{7} x^{2}) - \log_{2} (9)$

Step 3.2

Multiply $x^{7}$ by $x^{2}$ by adding the exponents.

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Step 3.2.1

Move $x^{2}$ .

$\log_{3} (9 (x^{2} x^{7})) - \log_{2} (9)$

Step 3.2.2

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

$\log_{3} (9 x^{2 + 7}) - \log_{2} (9)$

Step 3.2.3

Add $2$ and $7$ .

$\log_{3} (9 x^{9}) - \log_{2} (9)$