Precalculus Examples

$2 \log_{7} (x^{4}) - 6 \log_{7} (\sqrt[3]{x^{2}}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{7} (x^{4})$ by moving $2$ inside the logarithm.

$\log_{7} ({(x^{4})}^{2}) - 6 \log_{7} (\sqrt[3]{x^{2}}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.2

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 1.2.1

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

$\log_{7} (x^{4 \cdot 2}) - 6 \log_{7} (\sqrt[3]{x^{2}}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.2.2

Multiply $4$ by $2$ .

$\log_{7} (x^{8}) - 6 \log_{7} (\sqrt[3]{x^{2}}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.3

Simplify $- 6 \log_{7} (\sqrt[3]{x^{2}})$ by moving $6$ inside the logarithm.

$\log_{7} (x^{8}) - \log_{7} ({\sqrt[3]{x^{2}}}^{6}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.4

Rewrite ${\sqrt[3]{x^{2}}}^{6}$ as $x^{4}$ .

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Step 1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\log_{7} (x^{8}) - \log_{7} ({(x^{\frac{2}{3}})}^{6}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.4.2

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

$\log_{7} (x^{8}) - \log_{7} (x^{\frac{2}{3} \cdot 6}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.4.3

Combine $\frac{2}{3}$ and $6$ .

$\log_{7} (x^{8}) - \log_{7} (x^{\frac{2 \cdot 6}{3}}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.4.4

Multiply $2$ by $6$ .

$\log_{7} (x^{8}) - \log_{7} (x^{\frac{12}{3}}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.4.5

Cancel the common factor of $12$ and $3$ .

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Step 1.4.5.1

Factor $3$ out of $12$ .

$\log_{7} (x^{8}) - \log_{7} (x^{\frac{3 \cdot 4}{3}}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.4.5.2

Cancel the common factors.

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Step 1.4.5.2.1

Factor $3$ out of $3$ .

$\log_{7} (x^{8}) - \log_{7} (x^{\frac{3 \cdot 4}{3 (1)}}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.4.5.2.2

Cancel the common factor.

$\log_{7} (x^{8}) - \log_{7} (x^{\frac{3 \cdot 4}{3 \cdot 1}}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.4.5.2.3

Rewrite the expression.

$\log_{7} (x^{8}) - \log_{7} (x^{\frac{4}{1}}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.4.5.2.4

Divide $4$ by $1$ .

$\log_{7} (x^{8}) - \log_{7} (x^{4}) + 3 \log_{7} (4 x) + \log_{7} (2)$

Step 1.5

Simplify $3 \log_{7} (4 x)$ by moving $3$ inside the logarithm.

$\log_{7} (x^{8}) - \log_{7} (x^{4}) + \log_{7} ({(4 x)}^{3}) + \log_{7} (2)$

Step 1.6

Apply the product rule to $4 x$ .

$\log_{7} (x^{8}) - \log_{7} (x^{4}) + \log_{7} (4^{3} x^{3}) + \log_{7} (2)$

Step 1.7

Raise $4$ to the power of $3$ .

$\log_{7} (x^{8}) - \log_{7} (x^{4}) + \log_{7} (64 x^{3}) + \log_{7} (2)$

Step 2

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

$\log_{7} (\frac{x^{8}}{x^{4}}) + \log_{7} (64 x^{3}) + \log_{7} (2)$

Step 3

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

$\log_{7} (\frac{x^{8}}{x^{4}} (64 x^{3})) + \log_{7} (2)$

Step 4

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

$\log_{7} (\frac{x^{8}}{x^{4}} (64 x^{3}) \cdot 2)$

Step 5

Rewrite using the commutative property of multiplication.

$\log_{7} (64 \frac{x^{8}}{x^{4}} x^{3} \cdot 2)$

Step 6

Cancel the common factor of $x^{8}$ and $x^{4}$ .

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Step 6.1

Factor $x^{4}$ out of $x^{8}$ .

$\log_{7} (64 \frac{x^{4} x^{4}}{x^{4}} x^{3} \cdot 2)$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Multiply by $1$ .

$\log_{7} (64 \frac{x^{4} x^{4}}{x^{4} \cdot 1} x^{3} \cdot 2)$

Step 6.2.2

Cancel the common factor.

$\log_{7} (64 \frac{x^{4} x^{4}}{x^{4} \cdot 1} x^{3} \cdot 2)$

Step 6.2.3

Rewrite the expression.

$\log_{7} (64 \frac{x^{4}}{1} x^{3} \cdot 2)$

Step 6.2.4

Divide $x^{4}$ by $1$ .

$\log_{7} (64 x^{4} x^{3} \cdot 2)$

Step 7

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 7.1

Move $x^{3}$ .

$\log_{7} (64 (x^{3} x^{4}) \cdot 2)$

Step 7.2

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

$\log_{7} (64 x^{3 + 4} \cdot 2)$

Step 7.3

Add $3$ and $4$ .

$\log_{7} (64 x^{7} \cdot 2)$

Step 8

Multiply $2$ by $64$ .

$\log_{7} (128 x^{7})$