Precalculus Examples

$24 \log_{3} (\sqrt[3]{x}) + \log_{3} (24 x^{6}) - \log_{3} (24)$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$24 \log_{3} (\sqrt[3]{x}) + \log_{3} (\frac{24 x^{6}}{24})$

Step 2

Simplify each term.

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Step 2.1

Simplify $24 \log_{3} (\sqrt[3]{x})$ by moving $24$ inside the logarithm.

$\log_{3} ({\sqrt[3]{x}}^{24}) + \log_{3} (\frac{24 x^{6}}{24})$

Step 2.2

Rewrite ${\sqrt[3]{x}}^{24}$ as $x^{8}$ .

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Step 2.2.1

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

$\log_{3} ({(x^{\frac{1}{3}})}^{24}) + \log_{3} (\frac{24 x^{6}}{24})$

Step 2.2.2

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

$\log_{3} (x^{\frac{1}{3} \cdot 24}) + \log_{3} (\frac{24 x^{6}}{24})$

Step 2.2.3

Combine $\frac{1}{3}$ and $24$ .

$\log_{3} (x^{\frac{24}{3}}) + \log_{3} (\frac{24 x^{6}}{24})$

Step 2.2.4

Cancel the common factor of $24$ and $3$ .

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Step 2.2.4.1

Factor $3$ out of $24$ .

$\log_{3} (x^{\frac{3 \cdot 8}{3}}) + \log_{3} (\frac{24 x^{6}}{24})$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $3$ out of $3$ .

$\log_{3} (x^{\frac{3 \cdot 8}{3 (1)}}) + \log_{3} (\frac{24 x^{6}}{24})$

Step 2.2.4.2.2

Cancel the common factor.

$\log_{3} (x^{\frac{3 \cdot 8}{3 \cdot 1}}) + \log_{3} (\frac{24 x^{6}}{24})$

Step 2.2.4.2.3

Rewrite the expression.

$\log_{3} (x^{\frac{8}{1}}) + \log_{3} (\frac{24 x^{6}}{24})$

Step 2.2.4.2.4

Divide $8$ by $1$ .

$\log_{3} (x^{8}) + \log_{3} (\frac{24 x^{6}}{24})$

Step 2.3

Cancel the common factor of $24$ .

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Step 2.3.1

Cancel the common factor.

$\log_{3} (x^{8}) + \log_{3} (\frac{24 x^{6}}{24})$

Step 2.3.2

Divide $x^{6}$ by $1$ .

$\log_{3} (x^{8}) + \log_{3} (x^{6})$

Step 3

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

$\log_{3} (x^{8} x^{6})$

Step 4

Multiply $x^{8}$ by $x^{6}$ by adding the exponents.

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Step 4.1

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

$\log_{3} (x^{8 + 6})$

Step 4.2

Add $8$ and $6$ .

$\log_{3} (x^{14})$