Precalculus Examples

$8 \log_{6} (\sqrt{4 x^{- 1}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 \log_{6} (\sqrt{4 \frac{1}{x}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.2

Combine $4$ and $\frac{1}{x}$ .

$8 \log_{6} (\sqrt{\frac{4}{x}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.3

Rewrite $\sqrt{\frac{4}{x}}$ as $\frac{\sqrt{4}}{\sqrt{x}}$ .

$8 \log_{6} (\frac{\sqrt{4}}{\sqrt{x}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.4.2

Pull terms out from under the radical, assuming positive real numbers.

$8 \log_{6} (\frac{2}{\sqrt{x}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.5

Multiply $\frac{2}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

Multiply $\frac{2}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$8 \log_{6} (\frac{2 \sqrt{x}}{\sqrt{x} \sqrt{x}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.6.2

Raise $\sqrt{x}$ to the power of $1$ .

$8 \log_{6} (\frac{2 \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.6.3

Raise $\sqrt{x}$ to the power of $1$ .

$8 \log_{6} (\frac{2 \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.6.4

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

$8 \log_{6} (\frac{2 \sqrt{x}}{{\sqrt{x}}^{1 + 1}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.6.5

Add $1$ and $1$ .

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

Step 1.6.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 1.6.6.1

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

$8 \log_{6} (\frac{2 \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.6.6.2

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

$8 \log_{6} (\frac{2 \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.6.6.3

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

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

Step 1.6.6.4

Cancel the common factor of $2$ .

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Step 1.6.6.4.1

Cancel the common factor.

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

Step 1.6.6.4.2

Rewrite the expression.

$8 \log_{6} (\frac{2 \sqrt{x}}{x^{1}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.6.6.5

Simplify.

$8 \log_{6} (\frac{2 \sqrt{x}}{x}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.7

Simplify $8 \log_{6} (\frac{2 \sqrt{x}}{x})$ by moving $8$ inside the logarithm.

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

Step 1.8

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.8.1

Apply the product rule to $\frac{2 \sqrt{x}}{x}$ .

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

Step 1.8.2

Apply the product rule to $2 \sqrt{x}$ .

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

Step 1.9

Simplify the numerator.

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Step 1.9.1

Raise $2$ to the power of $8$ .

$\log_{6} (\frac{256 {\sqrt{x}}^{8}}{x^{8}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.9.2

Rewrite ${\sqrt{x}}^{8}$ as $x^{4}$ .

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Step 1.9.2.1

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

$\log_{6} (\frac{256 {(x^{\frac{1}{2}})}^{8}}{x^{8}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.9.2.2

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

$\log_{6} (\frac{256 x^{\frac{1}{2} \cdot 8}}{x^{8}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.9.2.3

Combine $\frac{1}{2}$ and $8$ .

$\log_{6} (\frac{256 x^{\frac{8}{2}}}{x^{8}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.9.2.4

Cancel the common factor of $8$ and $2$ .

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Step 1.9.2.4.1

Factor $2$ out of $8$ .

$\log_{6} (\frac{256 x^{\frac{2 \cdot 4}{2}}}{x^{8}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.9.2.4.2

Cancel the common factors.

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Step 1.9.2.4.2.1

Factor $2$ out of $2$ .

$\log_{6} (\frac{256 x^{\frac{2 \cdot 4}{2 (1)}}}{x^{8}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.9.2.4.2.2

Cancel the common factor.

$\log_{6} (\frac{256 x^{\frac{2 \cdot 4}{2 \cdot 1}}}{x^{8}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.9.2.4.2.3

Rewrite the expression.

$\log_{6} (\frac{256 x^{\frac{4}{1}}}{x^{8}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.9.2.4.2.4

Divide $4$ by $1$ .

$\log_{6} (\frac{256 x^{4}}{x^{8}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.10

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

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Step 1.10.1

Factor $x^{4}$ out of $256 x^{4}$ .

$\log_{6} (\frac{x^{4} \cdot 256}{x^{8}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.10.2

Cancel the common factors.

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Step 1.10.2.1

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

$\log_{6} (\frac{x^{4} \cdot 256}{x^{4} x^{4}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.10.2.2

Cancel the common factor.

$\log_{6} (\frac{x^{4} \cdot 256}{x^{4} x^{4}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 1.10.2.3

Rewrite the expression.

$\log_{6} (\frac{256}{x^{4}}) - \log_{6} (\frac{7}{x}) + \log_{6} (7)$

Step 2

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

$\log_{6} (\frac{\frac{256}{x^{4}}}{\frac{7}{x}}) + \log_{6} (7)$

Step 3

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

$\log_{6} (\frac{\frac{256}{x^{4}}}{\frac{7}{x}} \cdot 7)$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{6} (\frac{256}{x^{4}} \cdot \frac{x}{7} \cdot 7)$

Step 5

Combine.

$\log_{6} (\frac{256 x}{x^{4} \cdot 7} \cdot 7)$

Step 6

Cancel the common factor of $7$ .

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Step 6.1

Factor $7$ out of $x^{4} \cdot 7$ .

$\log_{6} (\frac{256 x}{7 \cdot x^{4}} \cdot 7)$

Step 6.2

Cancel the common factor.

$\log_{6} (\frac{256 x}{7 \cdot x^{4}} \cdot 7)$

Step 6.3

Rewrite the expression.

$\log_{6} (\frac{256 x}{x^{4}})$

Step 7

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

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Step 7.1

Factor $x$ out of $256 x$ .

$\log_{6} (\frac{x \cdot 256}{x^{4}})$

Step 7.2

Cancel the common factors.

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Step 7.2.1

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

$\log_{6} (\frac{x \cdot 256}{x \cdot x^{3}})$

Step 7.2.2

Cancel the common factor.

$\log_{6} (\frac{x \cdot 256}{x \cdot x^{3}})$

Step 7.2.3

Rewrite the expression.

$\log_{6} (\frac{256}{x^{3}})$